{"id":1065,"date":"2021-08-18T15:53:29","date_gmt":"2021-08-18T15:53:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1065"},"modified":"2022-01-26T22:49:06","modified_gmt":"2022-01-26T22:49:06","slug":"translate-verbal-statements-to-mathematical-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/translate-verbal-statements-to-mathematical-expressions\/","title":{"raw":"Translate Verbal Statements to Mathematical Expressions","rendered":"Translate Verbal Statements to Mathematical Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Express verbal statements as mathematical expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Translate word problems into expressions<\/h2>\r\nTo solve a word problem, you often need to translate a verbal expression into a mathematical expression. In the following table are some key words which correspond to common mathematical operations and relations.\r\n<div align=\"left\">\r\n<table style=\"height: 130px;\" width=\"607\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 123.391px;\">Addition<\/td>\r\n<td style=\"width: 15.75px;\">+<\/td>\r\n<td style=\"width: 429.359px;\">sum, total, together, more than, increased by, ...<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.391px;\">Subtraction<\/td>\r\n<td style=\"width: 15.75px;\">-<\/td>\r\n<td style=\"width: 429.359px;\">difference, less than, decreased by, ...<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.391px;\">Multiplication<\/td>\r\n<td style=\"width: 15.75px;\">*<\/td>\r\n<td style=\"width: 429.359px;\">product, double, half, percent of, ...<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 123.391px;\">Equality<\/td>\r\n<td style=\"width: 15.75px;\">=<\/td>\r\n<td style=\"width: 429.359px;\">is, is the same as, is not different from, ...<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<div align=\"left\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<h4>Verbal Statement<\/h4>\r\n<\/td>\r\n<td>\r\n<h4>Mathematical Statement<\/h4>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of a number and 5 is 12<\/td>\r\n<td>n+5=12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A number decreased by 5 is 12<\/td>\r\n<td>n-5=12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Half of a number is 12<\/td>\r\n<td>12n=12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The total of a number and 7 is the same as twice the number<\/td>\r\n<td>n+7=2n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\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\nFor more examples of how to translate verbal statements into algebraic statements, watch the following video.\r\n\r\nhttps:\/\/youtu.be\/Hub7ku7UHT4\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]142722[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Translate verbal statements involving inequalities into expressions<\/h2>\r\nOften we are interested in how a quantity compares to a particular value. For example, if more than 50% of voters vote in favor of a proposal, the proposal passes and is adopted. If we let [latex]p[\/latex] represent the proportion of voters who will vote in favor of a proposal, we can express the outcome \"the proposal passes\" as \"[latex]p[\/latex]\u00a0&gt; 0.5.\" Any value of [latex]p[\/latex]\u00a0which is more than 0.5 satisfies this inequality.\r\n\r\nThe table below shows the relationships between possible verbal statements about the relationship between a variable [latex]x[\/latex]\u00a0and a constant [latex]k[\/latex], and the corresponding mathematical statement.\r\n<div align=\"left\">\r\n<table style=\"height: 229px;\" width=\"590\">\r\n<tbody>\r\n<tr style=\"height: 73px;\">\r\n<td style=\"width: 323.125px; height: 73px;\">\r\n<h4>Verbal Statement: <em>x<\/em>\u00a0is...<\/h4>\r\n<\/td>\r\n<td style=\"width: 240.875px; height: 73px;\">\r\n<h4>Mathematical Statement<\/h4>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 323.125px; height: 26px;\">greater than or equal to [latex]k[\/latex]\r\nat least [latex]k[\/latex]\r\nnot more than [latex]k[\/latex]<\/td>\r\n<td style=\"width: 240.875px; height: 26px;\">\u00a0[latex]x \\geq k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 323.125px; height: 26px;\">less than or equal to [latex]k[\/latex]\r\nat most [latex]k[\/latex]\r\nnot more than [latex]k[\/latex]<\/td>\r\n<td style=\"width: 240.875px; height: 26px;\">\u00a0[latex]x \\leq k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 323.125px; height: 26px;\">less than [latex]k[\/latex]\r\nbelow [latex]k[\/latex]\r\nfewer than [latex]k[\/latex]<\/td>\r\n<td style=\"width: 240.875px; height: 26px;\">\u00a0[latex]x&lt;k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 323.125px; height: 26px;\">greater than [latex]k[\/latex]\r\nmore than [latex]k[\/latex]\r\nabove [latex]k[\/latex]<\/td>\r\n<td style=\"width: 240.875px; height: 26px;\">[latex]x&gt;k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 323.125px; height: 26px;\">equal to [latex]k[\/latex]\r\nis [latex]k[\/latex]\r\nexactly [latex]k[\/latex]\r\nthe same as [latex]k[\/latex]<\/td>\r\n<td style=\"width: 240.875px; height: 26px;\">[latex]x=k[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 26px;\">\r\n<td style=\"width: 323.125px; height: 26px;\">not equal to [latex]k[\/latex]\r\nnot\u00a0[latex]k[\/latex]\r\ndifferent from\u00a0[latex]k[\/latex]<\/td>\r\n<td style=\"width: 240.875px; height: 26px;\">[latex]x \\neq k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE: EXPRESS A VERBAL STATEMENT INVOLVING INEQUALITY AS A MATHEMATICAL STATEMENT<\/h3>\r\nWrite the following as a mathematical statement, using [latex]x[\/latex] for the variable.\r\n\r\nThe number of tries is no more than 12.\r\n\r\n[reveal-answer q=\"114880\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"114880\"]\r\n\r\nHere we are asked to use [latex]x[\/latex] for the unknown number of tries. The statement compares [latex]x[\/latex] to 12. Since the comparison is \"no more than,\" we use the symbol [latex]\\leq[\/latex].\r\n<p style=\"text-align: center;\">[latex]x \\leq 12[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE: EXPRESS A VERBAL STATEMENT INVOLVING INEQUALITY AS A MATHEMATICAL STATEMENT<\/h3>\r\nWrite the following as a mathematical statement, using [latex]x[\/latex]\u00a0for the variable.\r\n\r\nThe number of hearts is fewer than 5.\r\n\r\n[reveal-answer q=\"902367\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"902367\"]\r\n\r\nHere we are asked to use [latex]x[\/latex]\u00a0for the unknown number of tries. The statement compares [latex]x[\/latex] to 5.\u00a0 Since the comparison is \"fewer than,\" we use the symbol [latex]&lt;[\/latex].\r\n<p style=\"text-align: center;\">[latex]x &lt; 5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSome verbal statements involve compound inequalities. That is, two inequalities must both be satisfied.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE: EXPRESS A VERBAL STATEMENT INVOLVING INEQUALITY AS A MATHEMATICAL STATEMENT<\/h3>\r\nWrite the following as a mathematical statement, using [latex]x[\/latex] for the variable.\r\n\r\nThe number of defects is at least 1, but no more than 5.\r\n\r\n[reveal-answer q=\"669420\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"669420\"]\r\n\r\nHere we are asked to use [latex]x[\/latex] for the number of defects. The statement compares [latex]x[\/latex]\u00a0to both 1 and 5.\r\n\r\nSince the first comparison is \"at least,\" we use [latex]\\geq[\/latex]. So we require [latex]x\\geq1[\/latex].\r\n\r\nSince the second comparison is \"no more than,\" we use [latex]\\leq[\/latex]. So we require [latex]x\\leq5[\/latex].\r\n\r\nIn order for both comparisons to be true, we need [latex]x\\geq1[\/latex] and [latex]x\\leq5[\/latex]. Since we can interchange the expressions on each side of the inequality as long as we reverse the inequality, [latex]x\\geq1[\/latex] can also be written [latex]1\\leq x[\/latex]. We can write a single mathematical statement which describes the values of <em>x<\/em> which satisfy both conditions.\r\n<p style=\"text-align: center;\">[latex]1\\leq x \\leq 5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Express verbal statements as mathematical expressions<\/li>\n<\/ul>\n<\/div>\n<h2>Translate word problems into expressions<\/h2>\n<p>To solve a word problem, you often need to translate a verbal expression into a mathematical expression. In the following table are some key words which correspond to common mathematical operations and relations.<\/p>\n<div style=\"text-align: left;\">\n<table style=\"height: 130px; width: 607px;\">\n<tbody>\n<tr>\n<td style=\"width: 123.391px;\">Addition<\/td>\n<td style=\"width: 15.75px;\">+<\/td>\n<td style=\"width: 429.359px;\">sum, total, together, more than, increased by, &#8230;<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 123.391px;\">Subtraction<\/td>\n<td style=\"width: 15.75px;\">&#8211;<\/td>\n<td style=\"width: 429.359px;\">difference, less than, decreased by, &#8230;<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 123.391px;\">Multiplication<\/td>\n<td style=\"width: 15.75px;\">*<\/td>\n<td style=\"width: 429.359px;\">product, double, half, percent of, &#8230;<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 123.391px;\">Equality<\/td>\n<td style=\"width: 15.75px;\">=<\/td>\n<td style=\"width: 429.359px;\">is, is the same as, is not different from, &#8230;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<div style=\"text-align: left;\">\n<table>\n<tbody>\n<tr>\n<td>\n<h4>Verbal Statement<\/h4>\n<\/td>\n<td>\n<h4>Mathematical Statement<\/h4>\n<\/td>\n<\/tr>\n<tr>\n<td>The sum of a number and 5 is 12<\/td>\n<td>n+5=12<\/td>\n<\/tr>\n<tr>\n<td>A number decreased by 5 is 12<\/td>\n<td>n-5=12<\/td>\n<\/tr>\n<tr>\n<td>Half of a number is 12<\/td>\n<td>12n=12<\/td>\n<\/tr>\n<tr>\n<td>The total of a number and 7 is the same as twice the number<\/td>\n<td>n+7=2n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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>For more examples of how to translate verbal statements into algebraic statements, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142722\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142722&theme=oea&iframe_resize_id=ohm142722&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Translate verbal statements involving inequalities into expressions<\/h2>\n<p>Often we are interested in how a quantity compares to a particular value. For example, if more than 50% of voters vote in favor of a proposal, the proposal passes and is adopted. If we let [latex]p[\/latex] represent the proportion of voters who will vote in favor of a proposal, we can express the outcome &#8220;the proposal passes&#8221; as &#8220;[latex]p[\/latex]\u00a0&gt; 0.5.&#8221; Any value of [latex]p[\/latex]\u00a0which is more than 0.5 satisfies this inequality.<\/p>\n<p>The table below shows the relationships between possible verbal statements about the relationship between a variable [latex]x[\/latex]\u00a0and a constant [latex]k[\/latex], and the corresponding mathematical statement.<\/p>\n<div style=\"text-align: left;\">\n<table style=\"height: 229px; width: 590px;\">\n<tbody>\n<tr style=\"height: 73px;\">\n<td style=\"width: 323.125px; height: 73px;\">\n<h4>Verbal Statement: <em>x<\/em>\u00a0is&#8230;<\/h4>\n<\/td>\n<td style=\"width: 240.875px; height: 73px;\">\n<h4>Mathematical Statement<\/h4>\n<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 323.125px; height: 26px;\">greater than or equal to [latex]k[\/latex]<br \/>\nat least [latex]k[\/latex]<br \/>\nnot more than [latex]k[\/latex]<\/td>\n<td style=\"width: 240.875px; height: 26px;\">\u00a0[latex]x \\geq k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 323.125px; height: 26px;\">less than or equal to [latex]k[\/latex]<br \/>\nat most [latex]k[\/latex]<br \/>\nnot more than [latex]k[\/latex]<\/td>\n<td style=\"width: 240.875px; height: 26px;\">\u00a0[latex]x \\leq k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 323.125px; height: 26px;\">less than [latex]k[\/latex]<br \/>\nbelow [latex]k[\/latex]<br \/>\nfewer than [latex]k[\/latex]<\/td>\n<td style=\"width: 240.875px; height: 26px;\">\u00a0[latex]x<k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 323.125px; height: 26px;\">greater than [latex]k[\/latex]<br \/>\nmore than [latex]k[\/latex]<br \/>\nabove [latex]k[\/latex]<\/td>\n<td style=\"width: 240.875px; height: 26px;\">[latex]x>k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 323.125px; height: 26px;\">equal to [latex]k[\/latex]<br \/>\nis [latex]k[\/latex]<br \/>\nexactly [latex]k[\/latex]<br \/>\nthe same as [latex]k[\/latex]<\/td>\n<td style=\"width: 240.875px; height: 26px;\">[latex]x=k[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 26px;\">\n<td style=\"width: 323.125px; height: 26px;\">not equal to [latex]k[\/latex]<br \/>\nnot\u00a0[latex]k[\/latex]<br \/>\ndifferent from\u00a0[latex]k[\/latex]<\/td>\n<td style=\"width: 240.875px; height: 26px;\">[latex]x \\neq k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE: EXPRESS A VERBAL STATEMENT INVOLVING INEQUALITY AS A MATHEMATICAL STATEMENT<\/h3>\n<p>Write the following as a mathematical statement, using [latex]x[\/latex] for the variable.<\/p>\n<p>The number of tries is no more than 12.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q114880\">Show Answer<\/span><\/p>\n<div id=\"q114880\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here we are asked to use [latex]x[\/latex] for the unknown number of tries. The statement compares [latex]x[\/latex] to 12. Since the comparison is &#8220;no more than,&#8221; we use the symbol [latex]\\leq[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x \\leq 12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE: EXPRESS A VERBAL STATEMENT INVOLVING INEQUALITY AS A MATHEMATICAL STATEMENT<\/h3>\n<p>Write the following as a mathematical statement, using [latex]x[\/latex]\u00a0for the variable.<\/p>\n<p>The number of hearts is fewer than 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q902367\">Show Answer<\/span><\/p>\n<div id=\"q902367\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here we are asked to use [latex]x[\/latex]\u00a0for the unknown number of tries. The statement compares [latex]x[\/latex] to 5.\u00a0 Since the comparison is &#8220;fewer than,&#8221; we use the symbol [latex]<[\/latex].\n\n\n<p style=\"text-align: center;\">[latex]x < 5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Some verbal statements involve compound inequalities. That is, two inequalities must both be satisfied.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE: EXPRESS A VERBAL STATEMENT INVOLVING INEQUALITY AS A MATHEMATICAL STATEMENT<\/h3>\n<p>Write the following as a mathematical statement, using [latex]x[\/latex] for the variable.<\/p>\n<p>The number of defects is at least 1, but no more than 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q669420\">Show Answer<\/span><\/p>\n<div id=\"q669420\" class=\"hidden-answer\" style=\"display: none\">\n<p>Here we are asked to use [latex]x[\/latex] for the number of defects. The statement compares [latex]x[\/latex]\u00a0to both 1 and 5.<\/p>\n<p>Since the first comparison is &#8220;at least,&#8221; we use [latex]\\geq[\/latex]. So we require [latex]x\\geq1[\/latex].<\/p>\n<p>Since the second comparison is &#8220;no more than,&#8221; we use [latex]\\leq[\/latex]. So we require [latex]x\\leq5[\/latex].<\/p>\n<p>In order for both comparisons to be true, we need [latex]x\\geq1[\/latex] and [latex]x\\leq5[\/latex]. Since we can interchange the expressions on each side of the inequality as long as we reverse the inequality, [latex]x\\geq1[\/latex] can also be written [latex]1\\leq x[\/latex]. We can write a single mathematical statement which describes the values of <em>x<\/em> which satisfy both conditions.<\/p>\n<p style=\"text-align: center;\">[latex]1\\leq x \\leq 5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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-1065\">\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>Write Algebraic Expressions from Statements: Form ax+b and a(x+b). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Hub7ku7UHT4\">https:\/\/youtu.be\/Hub7ku7UHT4<\/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 Algebraic Expressions. <strong>Provided by<\/strong>: James Sousa (Mathispower4u.com). <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><strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 142722. <strong>Authored by<\/strong>: Lumen Learning. <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>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\">https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/a>. <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>: Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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