{"id":367,"date":"2018-04-17T02:25:11","date_gmt":"2018-04-17T02:25:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=367"},"modified":"2024-04-26T22:05:26","modified_gmt":"2024-04-26T22:05:26","slug":"translating-words-into-algebraic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/translating-words-into-algebraic-equations\/","title":{"raw":"Translating Words Into Algebra","rendered":"Translating Words Into Algebra"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning outcome<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Translate simple word phrases into math notation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13057,&quot;3&quot;:[null,0],&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Translate simple word phrases into math notation<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nWe know there are many operation symbols that are used in algebra. Now, we\u2019ll translate word phrases into algebraic expressions and equations. The symbols and variables we\u2019ve talked about will help us do that. They are summarized below.\r\n<table id=\"eip-386\" summary=\".\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Operation<\/th>\r\n<th data-align=\"center\">Phrase<\/th>\r\n<th data-align=\"center\">Expression<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><strong>Addition<\/strong><\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] plus [latex]b[\/latex]\r\n\r\nthe sum of [latex]a[\/latex] and [latex]b[\/latex]\r\n\r\n[latex]a[\/latex] increased by [latex]b[\/latex]\r\n\r\n[latex]b[\/latex] more than [latex]a[\/latex]\r\n\r\nthe total of [latex]a[\/latex] and [latex]b[\/latex]\r\n\r\n[latex]b[\/latex] added to [latex]a[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]a+b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><strong>Subtraction<\/strong><\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] minus [latex]b[\/latex]\r\n\r\nthe difference of [latex]a[\/latex] and [latex]b[\/latex]\r\n\r\n[latex]b[\/latex] subtracted from [latex]a[\/latex]\r\n\r\n[latex]a[\/latex] decreased by [latex]b[\/latex]\r\n\r\n[latex]b[\/latex] less than [latex]a[\/latex]<\/td>\r\n<td data-align=\"center\" data-valign=\"top\">[latex]a-b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><strong>Multiplication<\/strong><\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] times [latex]b[\/latex]\r\n\r\nthe product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<td data-align=\"center\" data-valign=\"top\">[latex]a\\cdot b[\/latex] , [latex]ab[\/latex] , [latex]a\\left(b\\right)[\/latex] , [latex]\\left(a\\right)\\left(b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\" data-valign=\"top\"><strong>Division<\/strong><\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] divided by [latex]b[\/latex]\r\n\r\nthe quotient of [latex]a[\/latex] and [latex]b[\/latex]\r\n\r\nthe ratio of [latex]a[\/latex] and [latex]b[\/latex]\r\n\r\n[latex]b[\/latex] divided into [latex]a[\/latex]<\/td>\r\n<td data-align=\"center\" data-valign=\"top\">[latex]a\\div b[\/latex] , [latex]a\/b[\/latex] , [latex]\\frac{a}{b}[\/latex] , [latex]b\\overline{)a}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLook closely at these phrases using the four operations:\r\n<ul id=\"fs-id1171105561839\" data-bullet-style=\"bullet\">\r\n \t<li>the sum <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\r\n \t<li>the difference <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\r\n \t<li>the product <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\r\n \t<li>the quotient <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\r\n<\/ul>\r\nEach phrase tells you to operate on two numbers. Look for the words <strong><em data-effect=\"italics\">of<\/em><\/strong> and <strong><em data-effect=\"italics\">and<\/em><\/strong> to find the numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate each word phrase into an algebraic expression:\r\n\r\n1. The difference of [latex]20[\/latex] and [latex]4[\/latex]\r\n\r\n2. The quotient of [latex]10x[\/latex] and [latex]3[\/latex]\r\n\r\nSolution\r\n<p style=\"padding-left: 30px;\">1. The key word is <em data-effect=\"italics\">difference<\/em>, which tells us the operation is subtraction. Look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to find the numbers to subtract.<\/p>\r\n<p style=\"padding-left: 90px;\">The difference of [latex]20[\/latex] and [latex]4[\/latex]<\/p>\r\n<p style=\"padding-left: 90px;\">[latex]20[\/latex] minus [latex]4[\/latex]<\/p>\r\n<p style=\"padding-left: 90px;\">[latex]20-4[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">2. The key word is <em data-effect=\"italics\">quotient<\/em>, which tells us the operation is division.<\/p>\r\n<p style=\"padding-left: 90px;\">The quotient of [latex]10x[\/latex] and [latex]3[\/latex]<\/p>\r\n<p style=\"padding-left: 90px;\">divide [latex]10x[\/latex] by [latex]3[\/latex]<\/p>\r\n<p style=\"padding-left: 90px;\">[latex]10x\\div{3}[\/latex]<\/p>\r\n<p style=\"padding-left: 90px;\">This can also be written as [latex](10x)\/3[\/latex] or [latex]\\frac{10x}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146541[\/ohm_question]\r\n\r\n[ohm_question]143240[\/ohm_question]\r\n\r\n[ohm_question]143207[\/ohm_question]\r\n\r\n[ohm_question]146542[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate each word phrase into an algebraic expression:\r\n<ol>\r\n \t<li>How old will you be in eight years? Let's say your current age is [latex]y[\/latex].<\/li>\r\n \t<li>How old were you seven years ago? This is seven years less than your age now. Let's say your current age is [latex]9z[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"879313\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"879313\"]\r\n\r\nSolution:\r\n\r\n1. Eight more than [latex]y[\/latex]\r\n<p style=\"padding-left: 30px;\">The key words are <em data-effect=\"italics\">more than<\/em>. They tell us the operation is addition. <em data-effect=\"italics\">More than<\/em> means \"added to\".<\/p>\r\n<p style=\"padding-left: 30px;\">Eight more than [latex]y[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">Eight added to [latex]y[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]y+8[\/latex]<\/p>\r\n2. Seven less than [latex]9z[\/latex].\r\n<p style=\"padding-left: 30px;\">The key words are <em data-effect=\"italics\">less than<\/em>. They tell us the operation is subtraction. <em data-effect=\"italics\">Less than<\/em> means \"subtracted from\".<\/p>\r\n<p style=\"padding-left: 30px;\">Seven less than [latex]9z[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">Seven subtracted from [latex]9z[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]9z - 7[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144907[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate each word phrase into an algebraic expression:\r\n\r\n1. Five times the sum of [latex]m[\/latex] and [latex]n[\/latex]\r\n2. The sum of five times [latex]m[\/latex] and [latex]n[\/latex]\r\n[reveal-answer q=\"888823\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"888823\"]\r\n\r\nSolution\r\n<p style=\"padding-left: 30px;\">1. There are two operation words: <em data-effect=\"italics\">times<\/em> tells us to multiply and <em data-effect=\"italics\">sum<\/em> tells us to add. Because we are multiplying [latex]5[\/latex] times the sum, we need parentheses around the sum of [latex]m[\/latex] and [latex]n[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">five times the sum of [latex]m[\/latex] and [latex]n[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">[latex]5(m+n)[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">2. To take a sum, we look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to see what is being added. Here we are taking the sum <em data-effect=\"italics\">of<\/em> five times [latex]m[\/latex] and [latex]n[\/latex].<\/p>\r\n<p style=\"padding-left: 60px;\">the sum of five times [latex]m[\/latex] and [latex]n[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">[latex]5m+n[\/latex]<\/p>\r\nNotice how the use of parentheses changes the result. In part 1, we add first and in part 2, we multiply first.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144916[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the video below to better understand how to write algebraic expressions from statements.\r\n\r\nhttps:\/\/youtu.be\/Hub7ku7UHT4\r\n\r\nWe'll eventually apply our skills in algebra to solving equations in complex word problems. Usually start by translating a word phrase to an algebraic equation. Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean <em>equals<\/em>. Some words that translate to the equal sign are:\r\n<ul id=\"fs-id2494527\">\r\n \t<li>is equal to<\/li>\r\n \t<li>is the same as<\/li>\r\n \t<li>is<\/li>\r\n \t<li>gives<\/li>\r\n \t<li>was<\/li>\r\n \t<li>will be<\/li>\r\n<\/ul>\r\nIt may be helpful to put a box around the <em>equals<\/em> word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate the sentence into an algebraic equation: The sum of [latex]6[\/latex] and [latex]9[\/latex] is [latex]15[\/latex].\r\n\r\nSolution\r\nThe word <em>is<\/em> tells us the equal sign goes between 9 and 15.\r\n<table id=\"eip-id1168468614513\" class=\"unnumbered unstyled\" style=\"height: 90px;\" summary=\"The image is 2 columns. The left column shows instructions and the right column shows word expressions and expressions. The first line in the left column says \">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 422.46875px;\">Locate the \"equals\" word(s).<\/td>\r\n<td style=\"height: 48px; width: 202px;\" rowspan=\"2\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215934\/CNX_BMath_Figure_02_03_032-01.png\" alt=\"The sum of 6 and 9 is 15. The word &quot;is&quot; translates to the equals sign.\" width=\"202\" height=\"58\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px;\">\r\n<td style=\"height: 36px; width: 422.46875px;\">Write the = sign.<\/td>\r\n<td style=\"height: 36px; width: 0.015625px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 21px;\">\r\n<td style=\"height: 21px; width: 422.46875px;\">Translate the words to the left of the <em>equals<\/em> word into an algebraic expression.<\/td>\r\n<td style=\"height: 21px; width: 202px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215935\/CNX_BMath_Figure_02_03_032-02.png\" alt=\"What does 6 plus 9 equal?\" width=\"202\" height=\"19\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 21px;\">\r\n<td style=\"height: 21px; width: 422.46875px;\">Translate the words to the right of the <em>equals<\/em> word into an algebraic expression.<\/td>\r\n<td style=\"height: 21px; width: 202px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215936\/CNX_BMath_Figure_02_03_032-03.png\" alt=\"6 plus 9 equals 15\" width=\"202\" height=\"19\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146546[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate the sentence into an algebraic equation: Twice the difference of [latex]x[\/latex] and [latex]3[\/latex] gives [latex]18[\/latex].\r\n[reveal-answer q=\"79895\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"79895\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469837874\" class=\"unnumbered unstyled\" summary=\"The image shows the sentence \">\r\n<tbody>\r\n<tr>\r\n<td>Locate the \"equals\" word(s).<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215940\/CNX_BMath_Figure_02_03_029_img-01.png\" alt=\"Twice the difference of x and 3 gives 18. Gives is our equals word.\" width=\"325\" height=\"26\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Recognize the key words: <em>twice; difference of \u2026. and \u2026<\/em>.<\/td>\r\n<td><em>Twice<\/em> means two times.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215942\/CNX_BMath_Figure_02_03_029_img-02.png\" alt=\"Twice translates to 2. The difference of x and 3 translates to parentheses x minus 3. Gives translates to the equals sign. 18 translates to itself.\" width=\"325\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2(x-3)=18[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146549[\/ohm_question]\r\n\r\n[ohm_question]146550[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=GSZWYghQhbQ\r\n\r\nNow let's apply our understanding of translating words to algebra in a real world scenario.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nThe height of a rectangular window is [latex]6[\/latex] inches less than the width. Let [latex]w[\/latex] represent the width of the window. Write an expression for the height of the window.\r\n[reveal-answer q=\"704093\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"704093\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466033010\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Write a phrase about the height.<\/td>\r\n<td>[latex]6[\/latex] less than the width<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]w[\/latex] for the width.<\/td>\r\n<td>[latex]6[\/latex] less than [latex]w[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite 'less than' as 'subtracted from'.<\/td>\r\n<td>[latex]6[\/latex] subtracted from [latex]w[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate the phrase into algebra.<\/td>\r\n<td>[latex]w - 6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144917[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nBlanca has dimes and quarters in her purse. The number of dimes is [latex]2[\/latex] less than [latex]5[\/latex] times the number of quarters. Let [latex]q[\/latex] represent the number of quarters. Write an expression for the number of dimes.\r\n[reveal-answer q=\"365611\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"365611\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467234223\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Write a phrase about the number of dimes.<\/td>\r\n<td>two less than five times the number of quarters<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]q[\/latex] for the number of quarters.<\/td>\r\n<td>[latex]2[\/latex] less than five times [latex]q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate [latex]5[\/latex] <em data-effect=\"italics\">times<\/em> [latex]q[\/latex] .<\/td>\r\n<td>[latex]2[\/latex] less than [latex]5q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate the phrase into algebra.<\/td>\r\n<td>[latex]5q - 2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144918[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to write basic algebraic expressions from words, and simplify.\r\n\r\nhttps:\/\/youtu.be\/x6b-OIBKSks\r\n\r\nLet\u2019s practice translating sentences into algebraic equations and then solving them.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate and solve: Three more than [latex]x[\/latex] is equal to [latex]47[\/latex].\r\n[reveal-answer q=\"989526\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"989526\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469829072\" class=\"unnumbered unstyled\" summary=\"From the equation x plus 3 equals 47, subtract 3 from both sides. The left side simplifies to just x. The right side is 47 minus 3 to get 44. The equation becomes x equal to 44. Check by letting x be 44 in the original equation. Is 44 plus 3 equal to 47? Yes.\">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>Three more than <em>x<\/em> is equal to [latex]47[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Translate.<\/td>\r\n<td>[latex]x+3=47[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Subtract [latex]3[\/latex] from both sides of the equation.<\/td>\r\n<td>[latex]x+3\\color{red}{--3}=47\\color{red}{--3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]x=44[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We can check. Let [latex]x=44[\/latex] .<\/td>\r\n<td>[latex]x+3=47[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]44+3=47[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]47=47\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]x=\\text{}44[\/latex] is the solution.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146551[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nTranslate and solve: The difference of [latex]y[\/latex] and [latex]14[\/latex] is [latex]18[\/latex].\r\n[reveal-answer q=\"763260\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"763260\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469862237\" class=\"unnumbered unstyled\" summary=\"The image shows the sentence \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>The difference of [latex]y[\/latex] and [latex]14[\/latex] is [latex]18[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Translate.<\/td>\r\n<td>[latex]y-14=18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add [latex]14[\/latex] to both sides.<\/td>\r\n<td>[latex]y-14\\color{red}{+14}=18\\color{red}{+14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]y=32[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We can check. Let [latex]y=32[\/latex] .<\/td>\r\n<td>[latex]y--14=18[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]32--14=18[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18=18\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]y=32[\/latex] is the solution.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146552[\/ohm_question]\r\n\r\n[ohm_question]146386[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to translate an equation into\u00a0words and solve. Note that this is different from the written examples on this page because we start with the mathematical equation then translate it into words.\r\n\r\nhttps:\/\/youtu.be\/tubom5d5lxg\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nTranslate from algebra to words:\r\n<ol>\r\n \t<li>[latex]12+14[\/latex]<\/li>\r\n \t<li>[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/li>\r\n \t<li>[latex]64\\div 8[\/latex]<\/li>\r\n \t<li>[latex]x-y[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 15.7812px;\">\r\n<td style=\"height: 15.7812px;\">1.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12+14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12[\/latex] plus [latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">the sum of twelve and fourteen<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]30[\/latex] times [latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">the product of thirty and five<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]64\\div 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]64[\/latex] divided by [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">the quotient of sixty-four and eight<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 15.5625px;\">\r\n<td style=\"height: 15.5625px;\">4.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x-y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x[\/latex] minus [latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">the difference of [latex]x[\/latex] and [latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144651&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"330\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144652&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php? id=144653&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\nWhat if we are working with expressions that are not equal? An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that [latex]b[\/latex] is greater than [latex]a[\/latex], it means that [latex]b[\/latex] is to the right of [latex]a[\/latex] on the number line. We use the symbols [latex]\\text{&lt;}[\/latex] and [latex]\\text{&gt;}[\/latex] for inequalities.\r\n<p style=\"text-align: center;\">[latex]a&lt;b[\/latex] is read [latex]a[\/latex] is less than [latex]b[\/latex]\r\n[latex]a[\/latex] is to the left of [latex]b[\/latex] on the number line<\/p>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215726\/CNX_BMath_Figure_02_01_001.png\" alt=\"The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.\" data-media-type=\"image\/png\" \/>\r\n[latex]a&gt;b[\/latex] is read [latex]a[\/latex] is greater than [latex]b[\/latex]\r\n[latex]a[\/latex] is to the right of [latex]b[\/latex] on the number line<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215727\/CNX_BMath_Figure_02_01_002.png\" alt=\"The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.\" data-media-type=\"image\/png\" \/>\r\n\r\nThe expressions [latex]a&lt;b\\text{ and }a&gt;b[\/latex] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a&lt;b\\text{ is equivalent to }b&gt;a.\\text{ For example, }7&lt;11\\text{ is equivalent to }11&gt;7.\\hfill \\\\ a&gt;b\\text{ is equivalent to }b&lt;a.\\text{ For example, }17&gt;4\\text{ is equivalent to }4&lt;17.\\hfill \\end{array}[\/latex]<\/p>\r\nWhen we write an inequality symbol with a line under it, such as [latex]a\\le b[\/latex], it means [latex]a&lt;b[\/latex] or [latex]a=b[\/latex]. We read this [latex]a[\/latex] is less than or equal to [latex]b[\/latex]. Also, if we put a slash through an equal sign, [latex]\\ne[\/latex], it means not equal.\r\n\r\nWe summarize the symbols of equality and inequality in the table below.\r\n<table style=\"width: 40%;\" summary=\"This table has seven rows and two columns. The first row is a header row and it labels each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Algebraic Notation<\/th>\r\n<th data-align=\"center\">Say<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]a=b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] is equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]a\\ne b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] is not equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]a&lt;b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] is less than [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]a&gt;b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] is greater than [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]a\\le b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] is less than or equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]a\\ge b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] is greater than or equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: left;\">Symbols [latex]&amp;lt[\/latex] and [latex]&amp;gt[\/latex]<\/h3>\r\n<p style=\"text-align: left;\">The symbols [latex]&amp;lt[\/latex] and [latex]&amp;gt[\/latex] each have a smaller side and a larger side.<\/p>\r\n<p style=\"text-align: center;\">smaller side [latex]&amp;lt[\/latex] larger side<\/p>\r\n<p style=\"text-align: center;\">larger side [latex]&amp;gt[\/latex] smaller side<\/p>\r\nThe smaller side of the symbol faces the smaller number and the larger faces the larger number.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nTranslate from algebra to words:\r\n<ol>\r\n \t<li>[latex]20\\le 35[\/latex]<\/li>\r\n \t<li>[latex]11\\ne 15 - 3[\/latex]<\/li>\r\n \t<li>[latex]9&gt;10\\div 2[\/latex]<\/li>\r\n \t<li>[latex]x+2&lt;10[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"346424\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"346424\"]\r\n\r\nSolution:\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]20\\le 35[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]20[\/latex] is less than or equal to [latex]35[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11\\ne 15 - 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11[\/latex] is not equal to [latex]15[\/latex] minus [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9&gt;10\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9[\/latex] is greater than [latex]10[\/latex] divided by [latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x+2&lt;10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex] plus [latex]2[\/latex] is less than [latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php? id=144653&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144654&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144655&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">In the following video we show more examples of how to write inequalities as words.<\/span>\r\n\r\nhttps:\/\/youtu.be\/q2ciQBwkjbk","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning outcome<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Translate simple word phrases into math notation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13057,&quot;3&quot;:[null,0],&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Translate simple word phrases into math notation<\/span><\/li>\n<\/ul>\n<\/div>\n<p>We know there are many operation symbols that are used in algebra. Now, we\u2019ll translate word phrases into algebraic expressions and equations. The symbols and variables we\u2019ve talked about will help us do that. They are summarized below.<\/p>\n<table id=\"eip-386\" summary=\".\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Operation<\/th>\n<th data-align=\"center\">Phrase<\/th>\n<th data-align=\"center\">Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><strong>Addition<\/strong><\/td>\n<td data-align=\"left\">[latex]a[\/latex] plus [latex]b[\/latex]<\/p>\n<p>the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/p>\n<p>[latex]a[\/latex] increased by [latex]b[\/latex]<\/p>\n<p>[latex]b[\/latex] more than [latex]a[\/latex]<\/p>\n<p>the total of [latex]a[\/latex] and [latex]b[\/latex]<\/p>\n<p>[latex]b[\/latex] added to [latex]a[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]a+b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><strong>Subtraction<\/strong><\/td>\n<td data-align=\"left\">[latex]a[\/latex] minus [latex]b[\/latex]<\/p>\n<p>the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/p>\n<p>[latex]b[\/latex] subtracted from [latex]a[\/latex]<\/p>\n<p>[latex]a[\/latex] decreased by [latex]b[\/latex]<\/p>\n<p>[latex]b[\/latex] less than [latex]a[\/latex]<\/td>\n<td data-align=\"center\" data-valign=\"top\">[latex]a-b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><strong>Multiplication<\/strong><\/td>\n<td data-align=\"left\">[latex]a[\/latex] times [latex]b[\/latex]<\/p>\n<p>the product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<td data-align=\"center\" data-valign=\"top\">[latex]a\\cdot b[\/latex] , [latex]ab[\/latex] , [latex]a\\left(b\\right)[\/latex] , [latex]\\left(a\\right)\\left(b\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\" data-valign=\"top\"><strong>Division<\/strong><\/td>\n<td data-align=\"left\">[latex]a[\/latex] divided by [latex]b[\/latex]<\/p>\n<p>the quotient of [latex]a[\/latex] and [latex]b[\/latex]<\/p>\n<p>the ratio of [latex]a[\/latex] and [latex]b[\/latex]<\/p>\n<p>[latex]b[\/latex] divided into [latex]a[\/latex]<\/td>\n<td data-align=\"center\" data-valign=\"top\">[latex]a\\div b[\/latex] , [latex]a\/b[\/latex] , [latex]\\frac{a}{b}[\/latex] , [latex]b\\overline{)a}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Look closely at these phrases using the four operations:<\/p>\n<ul id=\"fs-id1171105561839\" data-bullet-style=\"bullet\">\n<li>the sum <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\n<li>the difference <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\n<li>the product <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\n<li>the quotient <em data-effect=\"italics\">of<\/em> [latex]a[\/latex] <em data-effect=\"italics\">and<\/em> [latex]b[\/latex]<\/li>\n<\/ul>\n<p>Each phrase tells you to operate on two numbers. Look for the words <strong><em data-effect=\"italics\">of<\/em><\/strong> and <strong><em data-effect=\"italics\">and<\/em><\/strong> to find the numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate each word phrase into an algebraic expression:<\/p>\n<p>1. The difference of [latex]20[\/latex] and [latex]4[\/latex]<\/p>\n<p>2. The quotient of [latex]10x[\/latex] and [latex]3[\/latex]<\/p>\n<p>Solution<\/p>\n<p style=\"padding-left: 30px;\">1. The key word is <em data-effect=\"italics\">difference<\/em>, which tells us the operation is subtraction. Look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to find the numbers to subtract.<\/p>\n<p style=\"padding-left: 90px;\">The difference of [latex]20[\/latex] and [latex]4[\/latex]<\/p>\n<p style=\"padding-left: 90px;\">[latex]20[\/latex] minus [latex]4[\/latex]<\/p>\n<p style=\"padding-left: 90px;\">[latex]20-4[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">2. The key word is <em data-effect=\"italics\">quotient<\/em>, which tells us the operation is division.<\/p>\n<p style=\"padding-left: 90px;\">The quotient of [latex]10x[\/latex] and [latex]3[\/latex]<\/p>\n<p style=\"padding-left: 90px;\">divide [latex]10x[\/latex] by [latex]3[\/latex]<\/p>\n<p style=\"padding-left: 90px;\">[latex]10x\\div{3}[\/latex]<\/p>\n<p style=\"padding-left: 90px;\">This can also be written as [latex](10x)\/3[\/latex] or [latex]\\frac{10x}{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146541\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146541&theme=oea&iframe_resize_id=ohm146541&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm143240\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143240&theme=oea&iframe_resize_id=ohm143240&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm143207\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143207&theme=oea&iframe_resize_id=ohm143207&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146542\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146542&theme=oea&iframe_resize_id=ohm146542&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate each word phrase into an algebraic expression:<\/p>\n<ol>\n<li>How old will you be in eight years? Let&#8217;s say your current age is [latex]y[\/latex].<\/li>\n<li>How old were you seven years ago? This is seven years less than your age now. Let&#8217;s say your current age is [latex]9z[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q879313\">Show Answer<\/span><\/p>\n<div id=\"q879313\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<p>1. Eight more than [latex]y[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">The key words are <em data-effect=\"italics\">more than<\/em>. They tell us the operation is addition. <em data-effect=\"italics\">More than<\/em> means &#8220;added to&#8221;.<\/p>\n<p style=\"padding-left: 30px;\">Eight more than [latex]y[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">Eight added to [latex]y[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]y+8[\/latex]<\/p>\n<p>2. Seven less than [latex]9z[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">The key words are <em data-effect=\"italics\">less than<\/em>. They tell us the operation is subtraction. <em data-effect=\"italics\">Less than<\/em> means &#8220;subtracted from&#8221;.<\/p>\n<p style=\"padding-left: 30px;\">Seven less than [latex]9z[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">Seven subtracted from [latex]9z[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]9z - 7[\/latex]<\/p>\n<p style=\"padding-left: 30px;\"><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144907\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144907&theme=oea&iframe_resize_id=ohm144907&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate each word phrase into an algebraic expression:<\/p>\n<p>1. Five times the sum of [latex]m[\/latex] and [latex]n[\/latex]<br \/>\n2. The sum of five times [latex]m[\/latex] and [latex]n[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q888823\">Show Answer<\/span><\/p>\n<div id=\"q888823\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p style=\"padding-left: 30px;\">1. There are two operation words: <em data-effect=\"italics\">times<\/em> tells us to multiply and <em data-effect=\"italics\">sum<\/em> tells us to add. Because we are multiplying [latex]5[\/latex] times the sum, we need parentheses around the sum of [latex]m[\/latex] and [latex]n[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">five times the sum of [latex]m[\/latex] and [latex]n[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">[latex]5(m+n)[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">2. To take a sum, we look for the words <em data-effect=\"italics\">of<\/em> and <em data-effect=\"italics\">and<\/em> to see what is being added. Here we are taking the sum <em data-effect=\"italics\">of<\/em> five times [latex]m[\/latex] and [latex]n[\/latex].<\/p>\n<p style=\"padding-left: 60px;\">the sum of five times [latex]m[\/latex] and [latex]n[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">[latex]5m+n[\/latex]<\/p>\n<p>Notice how the use of parentheses changes the result. In part 1, we add first and in part 2, we multiply first.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144916\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144916&theme=oea&iframe_resize_id=ohm144916&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the video below to better understand how to write algebraic expressions from statements.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Write Algebraic Expressions from Statements: Form  ax+b and a(x+b)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hub7ku7UHT4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We&#8217;ll eventually apply our skills in algebra to solving equations in complex word problems. Usually start by translating a word phrase to an algebraic equation. Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean <em>equals<\/em>. Some words that translate to the equal sign are:<\/p>\n<ul id=\"fs-id2494527\">\n<li>is equal to<\/li>\n<li>is the same as<\/li>\n<li>is<\/li>\n<li>gives<\/li>\n<li>was<\/li>\n<li>will be<\/li>\n<\/ul>\n<p>It may be helpful to put a box around the <em>equals<\/em> word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate the sentence into an algebraic equation: The sum of [latex]6[\/latex] and [latex]9[\/latex] is [latex]15[\/latex].<\/p>\n<p>Solution<br \/>\nThe word <em>is<\/em> tells us the equal sign goes between 9 and 15.<\/p>\n<table id=\"eip-id1168468614513\" class=\"unnumbered unstyled\" style=\"height: 90px;\" summary=\"The image is 2 columns. The left column shows instructions and the right column shows word expressions and expressions. The first line in the left column says\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 422.46875px;\">Locate the &#8220;equals&#8221; word(s).<\/td>\n<td style=\"height: 48px; width: 202px;\" rowspan=\"2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215934\/CNX_BMath_Figure_02_03_032-01.png\" alt=\"The sum of 6 and 9 is 15. The word &quot;is&quot; translates to the equals sign.\" width=\"202\" height=\"58\" \/><\/td>\n<\/tr>\n<tr style=\"height: 36px;\">\n<td style=\"height: 36px; width: 422.46875px;\">Write the = sign.<\/td>\n<td style=\"height: 36px; width: 0.015625px;\"><\/td>\n<\/tr>\n<tr style=\"height: 21px;\">\n<td style=\"height: 21px; width: 422.46875px;\">Translate the words to the left of the <em>equals<\/em> word into an algebraic expression.<\/td>\n<td style=\"height: 21px; width: 202px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215935\/CNX_BMath_Figure_02_03_032-02.png\" alt=\"What does 6 plus 9 equal?\" width=\"202\" height=\"19\" \/><\/td>\n<\/tr>\n<tr style=\"height: 21px;\">\n<td style=\"height: 21px; width: 422.46875px;\">Translate the words to the right of the <em>equals<\/em> word into an algebraic expression.<\/td>\n<td style=\"height: 21px; width: 202px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215936\/CNX_BMath_Figure_02_03_032-03.png\" alt=\"6 plus 9 equals 15\" width=\"202\" height=\"19\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146546\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146546&theme=oea&iframe_resize_id=ohm146546&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate the sentence into an algebraic equation: Twice the difference of [latex]x[\/latex] and [latex]3[\/latex] gives [latex]18[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q79895\">Show Answer<\/span><\/p>\n<div id=\"q79895\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469837874\" class=\"unnumbered unstyled\" summary=\"The image shows the sentence\">\n<tbody>\n<tr>\n<td>Locate the &#8220;equals&#8221; word(s).<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215940\/CNX_BMath_Figure_02_03_029_img-01.png\" alt=\"Twice the difference of x and 3 gives 18. Gives is our equals word.\" width=\"325\" height=\"26\" \/><\/td>\n<\/tr>\n<tr>\n<td>Recognize the key words: <em>twice; difference of \u2026. and \u2026<\/em>.<\/td>\n<td><em>Twice<\/em> means two times.<\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215942\/CNX_BMath_Figure_02_03_029_img-02.png\" alt=\"Twice translates to 2. The difference of x and 3 translates to parentheses x minus 3. Gives translates to the equals sign. 18 translates to itself.\" width=\"325\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]2(x-3)=18[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146549\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146549&theme=oea&iframe_resize_id=ohm146549&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146550\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146550&theme=oea&iframe_resize_id=ohm146550&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Introduction to Variables and Variable Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/GSZWYghQhbQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now let&#8217;s apply our understanding of translating words to algebra in a real world scenario.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>The height of a rectangular window is [latex]6[\/latex] inches less than the width. Let [latex]w[\/latex] represent the width of the window. Write an expression for the height of the window.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q704093\">Show Answer<\/span><\/p>\n<div id=\"q704093\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466033010\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>Write a phrase about the height.<\/td>\n<td>[latex]6[\/latex] less than the width<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]w[\/latex] for the width.<\/td>\n<td>[latex]6[\/latex] less than [latex]w[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite &#8216;less than&#8217; as &#8216;subtracted from&#8217;.<\/td>\n<td>[latex]6[\/latex] subtracted from [latex]w[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate the phrase into algebra.<\/td>\n<td>[latex]w - 6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144917\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144917&theme=oea&iframe_resize_id=ohm144917&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Blanca has dimes and quarters in her purse. The number of dimes is [latex]2[\/latex] less than [latex]5[\/latex] times the number of quarters. Let [latex]q[\/latex] represent the number of quarters. Write an expression for the number of dimes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q365611\">Show Answer<\/span><\/p>\n<div id=\"q365611\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467234223\" class=\"unnumbered unstyled\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>Write a phrase about the number of dimes.<\/td>\n<td>two less than five times the number of quarters<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]q[\/latex] for the number of quarters.<\/td>\n<td>[latex]2[\/latex] less than five times [latex]q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate [latex]5[\/latex] <em data-effect=\"italics\">times<\/em> [latex]q[\/latex] .<\/td>\n<td>[latex]2[\/latex] less than [latex]5q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate the phrase into algebra.<\/td>\n<td>[latex]5q - 2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144918\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144918&theme=oea&iframe_resize_id=ohm144918&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to write basic algebraic expressions from words, and simplify.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Write Basic Expressions from Words Modeling Situations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/x6b-OIBKSks?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Let\u2019s practice translating sentences into algebraic equations and then solving them.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate and solve: Three more than [latex]x[\/latex] is equal to [latex]47[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q989526\">Show Answer<\/span><\/p>\n<div id=\"q989526\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469829072\" class=\"unnumbered unstyled\" summary=\"From the equation x plus 3 equals 47, subtract 3 from both sides. The left side simplifies to just x. The right side is 47 minus 3 to get 44. The equation becomes x equal to 44. Check by letting x be 44 in the original equation. Is 44 plus 3 equal to 47? Yes.\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>Three more than <em>x<\/em> is equal to [latex]47[\/latex].<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Translate.<\/td>\n<td>[latex]x+3=47[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Subtract [latex]3[\/latex] from both sides of the equation.<\/td>\n<td>[latex]x+3\\color{red}{--3}=47\\color{red}{--3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]x=44[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We can check. Let [latex]x=44[\/latex] .<\/td>\n<td>[latex]x+3=47[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]44+3=47[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]47=47\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]x=\\text{}44[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146551\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146551&theme=oea&iframe_resize_id=ohm146551&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Translate and solve: The difference of [latex]y[\/latex] and [latex]14[\/latex] is [latex]18[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q763260\">Show Answer<\/span><\/p>\n<div id=\"q763260\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469862237\" class=\"unnumbered unstyled\" summary=\"The image shows the sentence\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>The difference of [latex]y[\/latex] and [latex]14[\/latex] is [latex]18[\/latex].<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Translate.<\/td>\n<td>[latex]y-14=18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add [latex]14[\/latex] to both sides.<\/td>\n<td>[latex]y-14\\color{red}{+14}=18\\color{red}{+14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]y=32[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We can check. Let [latex]y=32[\/latex] .<\/td>\n<td>[latex]y--14=18[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]32--14=18[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18=18\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]y=32[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146552\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146552&theme=oea&iframe_resize_id=ohm146552&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146386\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146386&theme=oea&iframe_resize_id=ohm146386&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to translate an equation into\u00a0words and solve. Note that this is different from the written examples on this page because we start with the mathematical equation then translate it into words.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Solving Equations Using the Story of x (Part 1)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tubom5d5lxg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Translate from algebra to words:<\/p>\n<ol>\n<li>[latex]12+14[\/latex]<\/li>\n<li>[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/li>\n<li>[latex]64\\div 8[\/latex]<\/li>\n<li>[latex]x-y[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15.7812px;\">\n<td style=\"height: 15.7812px;\">1.<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12+14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12[\/latex] plus [latex]14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">the sum of twelve and fourteen<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]30[\/latex] times [latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">the product of thirty and five<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]64\\div 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]64[\/latex] divided by [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">the quotient of sixty-four and eight<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15.5625px;\">\n<td style=\"height: 15.5625px;\">4.<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x-y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x[\/latex] minus [latex]y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">the difference of [latex]x[\/latex] and [latex]y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144651&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"330\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144652&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php? id=144653&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<p>What if we are working with expressions that are not equal? An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that [latex]b[\/latex] is greater than [latex]a[\/latex], it means that [latex]b[\/latex] is to the right of [latex]a[\/latex] on the number line. We use the symbols [latex]\\text{<}[\/latex] and [latex]\\text{>}[\/latex] for inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]a<b[\/latex] is read [latex]a[\/latex] is less than [latex]b[\/latex]\n[latex]a[\/latex] is to the left of [latex]b[\/latex] on the number line<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215726\/CNX_BMath_Figure_02_01_001.png\" alt=\"The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.\" data-media-type=\"image\/png\" \/><br \/>\n[latex]a>b[\/latex] is read [latex]a[\/latex] is greater than [latex]b[\/latex]<br \/>\n[latex]a[\/latex] is to the right of [latex]b[\/latex] on the number line<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215727\/CNX_BMath_Figure_02_01_002.png\" alt=\"The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.\" data-media-type=\"image\/png\" \/><\/p>\n<p>The expressions [latex]a<b\\text{ and }a>b[\/latex] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a<b\\text{ is equivalent to }b>a.\\text{ For example, }7<11\\text{ is equivalent to }11>7.\\hfill \\\\ a>b\\text{ is equivalent to }b<a.\\text{ For example, }17>4\\text{ is equivalent to }4<17.\\hfill \\end{array}[\/latex]<\/p>\n<p>When we write an inequality symbol with a line under it, such as [latex]a\\le b[\/latex], it means [latex]a<b[\/latex] or [latex]a=b[\/latex]. We read this [latex]a[\/latex] is less than or equal to [latex]b[\/latex]. Also, if we put a slash through an equal sign, [latex]\\ne[\/latex], it means not equal.\n\nWe summarize the symbols of equality and inequality in the table below.\n\n\n<table style=\"width: 40%;\" summary=\"This table has seven rows and two columns. The first row is a header row and it labels each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Algebraic Notation<\/th>\n<th data-align=\"center\">Say<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]a=b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] is equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]a\\ne b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] is not equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]a<b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] is less than [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]a>b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] is greater than [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]a\\le b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] is less than or equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]a\\ge b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] is greater than or equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: left;\">Symbols [latex]&lt[\/latex] and [latex]&gt[\/latex]<\/h3>\n<p style=\"text-align: left;\">The symbols [latex]&lt[\/latex] and [latex]&gt[\/latex] each have a smaller side and a larger side.<\/p>\n<p style=\"text-align: center;\">smaller side [latex]&lt[\/latex] larger side<\/p>\n<p style=\"text-align: center;\">larger side [latex]&gt[\/latex] smaller side<\/p>\n<p>The smaller side of the symbol faces the smaller number and the larger faces the larger number.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Translate from algebra to words:<\/p>\n<ol>\n<li>[latex]20\\le 35[\/latex]<\/li>\n<li>[latex]11\\ne 15 - 3[\/latex]<\/li>\n<li>[latex]9>10\\div 2[\/latex]<\/li>\n<li>[latex]x+2<10[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q346424\">Show Answer<\/span><\/p>\n<div id=\"q346424\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>[latex]20\\le 35[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]20[\/latex] is less than or equal to [latex]35[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td>[latex]11\\ne 15 - 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]11[\/latex] is not equal to [latex]15[\/latex] minus [latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>[latex]9>10\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]9[\/latex] is greater than [latex]10[\/latex] divided by [latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<\/tr>\n<tr>\n<td>[latex]x+2<10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex] plus [latex]2[\/latex] is less than [latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php? id=144653&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144654&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144655&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #000000;\">In the following video we show more examples of how to write inequalities as words.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Write Inequalities as Words\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q2ciQBwkjbk?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-367\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"563a4a51-d6d9-48ee-afd9-40d3c8621591, 9ba782a8-6b59-4bb7-a68d-bf47b1c0ab04","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-367","chapter","type-chapter","status-publish","hentry"],"part":26,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/367","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":21,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/367\/revisions"}],"predecessor-version":[{"id":3996,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/367\/revisions\/3996"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/parts\/26"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/367\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=367"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=367"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=367"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}