{"id":4535,"date":"2017-06-07T18:50:33","date_gmt":"2017-06-07T18:50:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-define-a-process-for-problem-solving\/"},"modified":"2017-08-15T15:00:07","modified_gmt":"2017-08-15T15:00:07","slug":"read-define-a-process-for-problem-solving","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-define-a-process-for-problem-solving\/","title":{"raw":"Translating Sentences and Define a Process for Problem Solving","rendered":"Translating Sentences and Define a Process for Problem Solving"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Define a process for problem solving\r\n<ul>\r\n \t<li>Translate words into algebraic expressions and equations<\/li>\r\n \t<li>Define a process for solving word problems<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nWord problems can be tricky. Often it takes a bit of practice to convert an English sentence into a mathematical sentence, which is one of the first steps to solving word problems. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it's good to see what mathematical operators are associated with them.\r\n\r\nHow much will it cost?\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Addition [latex]+[\/latex]<\/th>\r\n<th>Subtraction [latex]-[\/latex]<\/th>\r\n<th>Multiplication [latex]\\times[\/latex]<\/th>\r\n<th>Variable ?<\/th>\r\n<th>Equals [latex]=[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>More than<\/td>\r\n<td>Less than<\/td>\r\n<td>Double<\/td>\r\n<td>A number<\/td>\r\n<td>Is<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Together<\/td>\r\n<td>In the past<\/td>\r\n<td>Product<\/td>\r\n<td>Often, a value for which no information is given.<\/td>\r\n<td>The same as<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Sum<\/td>\r\n<td>slower than<\/td>\r\n<td>\u00a0times<\/td>\r\n<td>After how many hours?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td>the remainder of<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>In the future<\/td>\r\n<td>\u00a0difference<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>faster than<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSome examples follow:\r\n<ul>\r\n \t<li>[latex]x\\text{ is }5[\/latex] \u00a0becomes [latex]x=5[\/latex]<\/li>\r\n \t<li>Three more than a number becomes [latex]x+3[\/latex]<\/li>\r\n \t<li>Four less than a number becomes [latex]x-4[\/latex]<\/li>\r\n \t<li>Double the cost becomes [latex]2\\cdot\\text{ cost }[\/latex]<\/li>\r\n \t<li>Groceries and gas together for the week cost $250 means [latex]\\text{ groceries }+\\text{ gas }=250[\/latex]<\/li>\r\n \t<li>The difference of 9 and a number becomes [latex]9-x[\/latex]. Notice how 9 is first in the sentence and the expression<\/li>\r\n<\/ul>\r\nLet's practice translating a few more English phrases into algebraic\u00a0expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate the table into algebraic expressions:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0some number<\/td>\r\n<td>\u00a0the sum of the number and 3<\/td>\r\n<td>\u00a0twice the sum of the number and 3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0a length<\/td>\r\n<td>\u00a0double the length<\/td>\r\n<td>\u00a0double the length, decreased by 6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0a cost<\/td>\r\n<td>\u00a0the difference of the cost and 20<\/td>\r\n<td>\u00a02 times the difference of the cost and 20<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0some quantity<\/td>\r\n<td>\u00a0the difference of 5 and the quantity<\/td>\r\n<td>\u00a0\u00a0the difference of 5 and the quantity, divided by 2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0an amount of time<\/td>\r\n<td>\u00a0triple the amount of time<\/td>\r\n<td>\u00a0triple the amount of time, increased by 5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0a distance<\/td>\r\n<td>\u00a0the sum of [latex]-4[\/latex] and the distance<\/td>\r\n<td>\u00a0the sum of [latex]-4[\/latex] and the twice the distance<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"790402\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"790402\"]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\u00a0[latex]a[\/latex]<\/td>\r\n<td>\u00a0[latex]a+3[\/latex]<\/td>\r\n<td>\u00a0[latex]2\\left(x+3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]l[\/latex]<\/td>\r\n<td>\u00a0[latex]2l[\/latex]<\/td>\r\n<td>\u00a0[latex]2l-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]c[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]c-20[\/latex]<\/td>\r\n<td>\u00a0[latex]2\\left(c-20\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]q[\/latex]<\/td>\r\n<td>\u00a0[latex]5-q[\/latex]<\/td>\r\n<td>\u00a0[latex]\\frac{5-q}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]t[\/latex]<\/td>\r\n<td>\u00a0[latex]3t[\/latex]<\/td>\r\n<td>\u00a0[latex]3t+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]d[\/latex]<\/td>\r\n<td>\u00a0[latex]-4+d[\/latex]<\/td>\r\n<td>\u00a0[latex]-4+2d[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this example video, we show how to translate more words into mathematical expressions.\r\n\r\nhttps:\/\/youtu.be\/uD_V5t-6Kzs\r\n\r\nThe power of algebra is how it can help you model real situations in order to answer questions about them.\r\n\r\nHere are some\u00a0steps to translate problem situations into algebraic equations you can solve. Not <em>every<\/em> word problem fits perfectly into these steps, but they will help you get started.\r\n<ol>\r\n \t<li>Read and understand the problem.<\/li>\r\n \t<li>Determine the constants and variables in the problem.<\/li>\r\n \t<li>Translate words into algebraic expressions and equations.<\/li>\r\n \t<li>Write an equation to represent the problem.<\/li>\r\n \t<li>Solve the equation.<\/li>\r\n \t<li>Check and interpret your answer. Sometimes writing a sentence helps.<\/li>\r\n<\/ol>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTwenty-eight\u00a0less than five times a certain number is 232. What is the number?\r\n\r\n[reveal-answer q=\"720402\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"720402\"]\r\n\r\nFollowing the steps provided:\r\n<ol>\r\n \t<li><strong>Read and understand:<\/strong> we are looking for a number.<\/li>\r\n \t<li><strong>Constants and variables:<\/strong> 28 and 232 are constants, \"a certain number\" is our variable because we don't know its value, and we are asked to find it. We will call it <em>x.<\/em><\/li>\r\n \t<li><strong>Translate:\u00a0<\/strong>five times a certain number translates to [latex]5x[\/latex]\r\nTwenty-eight\u00a0less than five times a certain number translates to\u00a0[latex]5x-28[\/latex] because subtraction is built backward.\r\nis 232 translates to [latex]=232[\/latex] because \"is\" is associated with equals.<\/li>\r\n \t<li><strong>Write an equation:<\/strong>\u00a0[latex]5x-28=232[\/latex]<\/li>\r\n \t<li><strong>Solve the equation using what you know about solving linear equations:<\/strong>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5x-28=232\\\\5x=260\\\\x=52\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<\/li>\r\n \t<li><strong>Check and interpret:<\/strong> We can substitute 52 for x.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5\\left(52\\right)-28=232\\\\5\\left(52\\right)=260\\\\260=260\\end{array}[\/latex].<\/p>\r\nTRUE!<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.\r\n\r\nhttps:\/\/youtu.be\/izIIqOztUyI\r\n\r\nAnother type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as 3, 4, 5. If we are looking for several consecutive numbers it is important to first identify what they look like with variables before we set up the equation.\r\n\r\nFor example, let's say I want to know the next consecutive integer after 4. In mathematical terms, we would add 1 to 4 to get 5. We can generalize this idea as follows: the consecutive integer of any number, <em>x<\/em>, is [latex]x+1[\/latex]. If we continue this pattern we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>First<\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Second<\/td>\r\n<td>[latex]x+1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Third<\/td>\r\n<td>[latex]x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Fourth<\/td>\r\n<td>\u00a0[latex]x+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe apply the idea of consecutive integers to solving a word problem in the following example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe sum of three consecutive integers is 93. What are the integers?\r\n\r\n[reveal-answer q=\"120402\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"120402\"]\r\nFollowing the steps provided:\r\n<ol>\r\n \t<li><strong>Read and understand:<\/strong>\u00a0We are looking for three numbers, and we know they are consecutive integers.<\/li>\r\n \t<li><strong>Constants and Variables:\u00a0<\/strong>93 is a constant.\r\nThe first integer we will call <em>x<\/em>.\r\nSecond: [latex]x+1[\/latex]\r\nThird: [latex]x+2[\/latex]<\/li>\r\n \t<li><strong>Translate:\u00a0<\/strong>The sum of three consecutive integers translates to [latex]x+\\left(x+1\\right)+\\left(x+2\\right)[\/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. <em>is 93<\/em> translates to [latex]=93[\/latex] because <em>is<\/em> is associated with equals.<\/li>\r\n \t<li><strong>Write an equation:<\/strong>\u00a0[latex]x+\\left(x+1\\right)+\\left(x+2\\right)=93[\/latex]<\/li>\r\n \t<li><strong>Solve the equation using what you know about solving linear equations:\u00a0<\/strong>We can't simplify within each set of parentheses, and we don't need to use the distributive property so we can rewrite the equation without parentheses.\r\n<p style=\"text-align: center\">[latex]x+x+1+x+2=93[\/latex]<\/p>\r\nCombine like terms, simplify, and solve.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}x+x+1+x+2=93\\\\3x+3 = 93\\\\\\underline{-3\\,\\,\\,\\,\\,-3}\\\\3x=90\\\\\\frac{3x}{3}=\\frac{90}{3}\\\\x=30\\end{array}[\/latex]<\/p>\r\n<\/li>\r\n \t<li><strong>Check and Interpret:<\/strong> Okay, we have found a value for <em>x<\/em>. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables: The first integer we will call [latex]x[\/latex], [latex]x=30[\/latex]\r\nSecond: [latex]x+1[\/latex] so [latex]30+1=31[\/latex]\r\nThird: [latex]x+2[\/latex] so [latex]30+2=32[\/latex] The three consecutive integers whose sum is [latex]93[\/latex] are [latex]30\\text{, }31\\text{, and }32[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of a consecutive integer problem.\r\nhttps:\/\/youtu.be\/S5HZy3jKodg","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Define a process for problem solving\n<ul>\n<li>Translate words into algebraic expressions and equations<\/li>\n<li>Define a process for solving word problems<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>Word problems can be tricky. Often it takes a bit of practice to convert an English sentence into a mathematical sentence, which is one of the first steps to solving word problems. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it&#8217;s good to see what mathematical operators are associated with them.<\/p>\n<p>How much will it cost?<\/p>\n<table>\n<thead>\n<tr>\n<th>Addition [latex]+[\/latex]<\/th>\n<th>Subtraction [latex]-[\/latex]<\/th>\n<th>Multiplication [latex]\\times[\/latex]<\/th>\n<th>Variable ?<\/th>\n<th>Equals [latex]=[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>More than<\/td>\n<td>Less than<\/td>\n<td>Double<\/td>\n<td>A number<\/td>\n<td>Is<\/td>\n<\/tr>\n<tr>\n<td>Together<\/td>\n<td>In the past<\/td>\n<td>Product<\/td>\n<td>Often, a value for which no information is given.<\/td>\n<td>The same as<\/td>\n<\/tr>\n<tr>\n<td>Sum<\/td>\n<td>slower than<\/td>\n<td>\u00a0times<\/td>\n<td>After how many hours?<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>the remainder of<\/td>\n<\/tr>\n<tr>\n<td>In the future<\/td>\n<td>\u00a0difference<\/td>\n<\/tr>\n<tr>\n<td>faster than<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Some examples follow:<\/p>\n<ul>\n<li>[latex]x\\text{ is }5[\/latex] \u00a0becomes [latex]x=5[\/latex]<\/li>\n<li>Three more than a number becomes [latex]x+3[\/latex]<\/li>\n<li>Four less than a number becomes [latex]x-4[\/latex]<\/li>\n<li>Double the cost becomes [latex]2\\cdot\\text{ cost }[\/latex]<\/li>\n<li>Groceries and gas together for the week cost $250 means [latex]\\text{ groceries }+\\text{ gas }=250[\/latex]<\/li>\n<li>The difference of 9 and a number becomes [latex]9-x[\/latex]. Notice how 9 is first in the sentence and the expression<\/li>\n<\/ul>\n<p>Let&#8217;s practice translating a few more English phrases into algebraic\u00a0expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate the table into algebraic expressions:<\/p>\n<table>\n<tbody>\n<tr>\n<td>\u00a0some number<\/td>\n<td>\u00a0the sum of the number and 3<\/td>\n<td>\u00a0twice the sum of the number and 3<\/td>\n<\/tr>\n<tr>\n<td>\u00a0a length<\/td>\n<td>\u00a0double the length<\/td>\n<td>\u00a0double the length, decreased by 6<\/td>\n<\/tr>\n<tr>\n<td>\u00a0a cost<\/td>\n<td>\u00a0the difference of the cost and 20<\/td>\n<td>\u00a02 times the difference of the cost and 20<\/td>\n<\/tr>\n<tr>\n<td>\u00a0some quantity<\/td>\n<td>\u00a0the difference of 5 and the quantity<\/td>\n<td>\u00a0\u00a0the difference of 5 and the quantity, divided by 2<\/td>\n<\/tr>\n<tr>\n<td>\u00a0an amount of time<\/td>\n<td>\u00a0triple the amount of time<\/td>\n<td>\u00a0triple the amount of time, increased by 5<\/td>\n<\/tr>\n<tr>\n<td>\u00a0a distance<\/td>\n<td>\u00a0the sum of [latex]-4[\/latex] and the distance<\/td>\n<td>\u00a0the sum of [latex]-4[\/latex] and the twice the distance<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q790402\">Show Solution<\/span><\/p>\n<div id=\"q790402\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>\u00a0[latex]a[\/latex]<\/td>\n<td>\u00a0[latex]a+3[\/latex]<\/td>\n<td>\u00a0[latex]2\\left(x+3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]l[\/latex]<\/td>\n<td>\u00a0[latex]2l[\/latex]<\/td>\n<td>\u00a0[latex]2l-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]c[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]c-20[\/latex]<\/td>\n<td>\u00a0[latex]2\\left(c-20\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]q[\/latex]<\/td>\n<td>\u00a0[latex]5-q[\/latex]<\/td>\n<td>\u00a0[latex]\\frac{5-q}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]t[\/latex]<\/td>\n<td>\u00a0[latex]3t[\/latex]<\/td>\n<td>\u00a0[latex]3t+5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]d[\/latex]<\/td>\n<td>\u00a0[latex]-4+d[\/latex]<\/td>\n<td>\u00a0[latex]-4+2d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In this example video, we show how to translate more words into mathematical expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Writing Algebraic Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/uD_V5t-6Kzs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The power of algebra is how it can help you model real situations in order to answer questions about them.<\/p>\n<p>Here are some\u00a0steps to translate problem situations into algebraic equations you can solve. Not <em>every<\/em> word problem fits perfectly into these steps, but they will help you get started.<\/p>\n<ol>\n<li>Read and understand the problem.<\/li>\n<li>Determine the constants and variables in the problem.<\/li>\n<li>Translate words into algebraic expressions and equations.<\/li>\n<li>Write an equation to represent the problem.<\/li>\n<li>Solve the equation.<\/li>\n<li>Check and interpret your answer. Sometimes writing a sentence helps.<\/li>\n<\/ol>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Twenty-eight\u00a0less than five times a certain number is 232. What is the number?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q720402\">Show Solution<\/span><\/p>\n<div id=\"q720402\" class=\"hidden-answer\" style=\"display: none\">\n<p>Following the steps provided:<\/p>\n<ol>\n<li><strong>Read and understand:<\/strong> we are looking for a number.<\/li>\n<li><strong>Constants and variables:<\/strong> 28 and 232 are constants, &#8220;a certain number&#8221; is our variable because we don&#8217;t know its value, and we are asked to find it. We will call it <em>x.<\/em><\/li>\n<li><strong>Translate:\u00a0<\/strong>five times a certain number translates to [latex]5x[\/latex]<br \/>\nTwenty-eight\u00a0less than five times a certain number translates to\u00a0[latex]5x-28[\/latex] because subtraction is built backward.<br \/>\nis 232 translates to [latex]=232[\/latex] because &#8220;is&#8221; is associated with equals.<\/li>\n<li><strong>Write an equation:<\/strong>\u00a0[latex]5x-28=232[\/latex]<\/li>\n<li><strong>Solve the equation using what you know about solving linear equations:<\/strong>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5x-28=232\\\\5x=260\\\\x=52\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/li>\n<li><strong>Check and interpret:<\/strong> We can substitute 52 for x.\n<p style=\"text-align: center\">[latex]\\begin{array}{r}5\\left(52\\right)-28=232\\\\5\\left(52\\right)=260\\\\260=260\\end{array}[\/latex].<\/p>\n<p>TRUE!<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Write and Solve a Linear Equations to Solve a Number Problem (1)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/izIIqOztUyI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as 3, 4, 5. If we are looking for several consecutive numbers it is important to first identify what they look like with variables before we set up the equation.<\/p>\n<p>For example, let&#8217;s say I want to know the next consecutive integer after 4. In mathematical terms, we would add 1 to 4 to get 5. We can generalize this idea as follows: the consecutive integer of any number, <em>x<\/em>, is [latex]x+1[\/latex]. If we continue this pattern we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.<\/p>\n<table>\n<tbody>\n<tr>\n<td>First<\/td>\n<td>[latex]x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Second<\/td>\n<td>[latex]x+1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Third<\/td>\n<td>[latex]x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Fourth<\/td>\n<td>\u00a0[latex]x+3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We apply the idea of consecutive integers to solving a word problem in the following example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The sum of three consecutive integers is 93. What are the integers?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q120402\">Show Solution<\/span><\/p>\n<div id=\"q120402\" class=\"hidden-answer\" style=\"display: none\">\nFollowing the steps provided:<\/p>\n<ol>\n<li><strong>Read and understand:<\/strong>\u00a0We are looking for three numbers, and we know they are consecutive integers.<\/li>\n<li><strong>Constants and Variables:\u00a0<\/strong>93 is a constant.<br \/>\nThe first integer we will call <em>x<\/em>.<br \/>\nSecond: [latex]x+1[\/latex]<br \/>\nThird: [latex]x+2[\/latex]<\/li>\n<li><strong>Translate:\u00a0<\/strong>The sum of three consecutive integers translates to [latex]x+\\left(x+1\\right)+\\left(x+2\\right)[\/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. <em>is 93<\/em> translates to [latex]=93[\/latex] because <em>is<\/em> is associated with equals.<\/li>\n<li><strong>Write an equation:<\/strong>\u00a0[latex]x+\\left(x+1\\right)+\\left(x+2\\right)=93[\/latex]<\/li>\n<li><strong>Solve the equation using what you know about solving linear equations:\u00a0<\/strong>We can&#8217;t simplify within each set of parentheses, and we don&#8217;t need to use the distributive property so we can rewrite the equation without parentheses.\n<p style=\"text-align: center\">[latex]x+x+1+x+2=93[\/latex]<\/p>\n<p>Combine like terms, simplify, and solve.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}x+x+1+x+2=93\\\\3x+3 = 93\\\\\\underline{-3\\,\\,\\,\\,\\,-3}\\\\3x=90\\\\\\frac{3x}{3}=\\frac{90}{3}\\\\x=30\\end{array}[\/latex]<\/p>\n<\/li>\n<li><strong>Check and Interpret:<\/strong> Okay, we have found a value for <em>x<\/em>. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables: The first integer we will call [latex]x[\/latex], [latex]x=30[\/latex]<br \/>\nSecond: [latex]x+1[\/latex] so [latex]30+1=31[\/latex]<br \/>\nThird: [latex]x+2[\/latex] so [latex]30+2=32[\/latex] The three consecutive integers whose sum is [latex]93[\/latex] are [latex]30\\text{, }31\\text{, and }32[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show another example of a consecutive integer problem.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Write and Solve a Linear Equations to Solve a Number Problem (Consecutive Integers)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/S5HZy3jKodg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4535\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Writing Algebraic Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/uD_V5t-6Kzs\">https:\/\/youtu.be\/uD_V5t-6Kzs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Write and Solve a Linear Equations to Solve a Number Problem (1). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/izIIqOztUyI\">https:\/\/youtu.be\/izIIqOztUyI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Write and Solve a Linear Equations to Solve a Number Problem (Consecutive Integers). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/S5HZy3jKodg\">https:\/\/youtu.be\/S5HZy3jKodg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Writing Algebraic Expressions\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/uD_V5t-6Kzs\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Write and Solve a Linear Equations to Solve a Number Problem (1)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/izIIqOztUyI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Write and Solve a Linear Equations to Solve a Number Problem (Consecutive Integers)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/S5HZy3jKodg\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"059a85db-6d19-450d-9cf4-7cce5f9e1e51","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4535","chapter","type-chapter","status-publish","hentry"],"part":4520,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4535\/revisions"}],"predecessor-version":[{"id":4993,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4535\/revisions\/4993"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/parts\/4520"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4535\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/media?parent=4535"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapter-type?post=4535"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/contributor?post=4535"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/license?post=4535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}