{"id":96,"date":"2023-06-21T13:22:32","date_gmt":"2023-06-21T13:22:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/temp-4-review\/"},"modified":"2024-01-08T18:47:50","modified_gmt":"2024-01-08T18:47:50","slug":"temp-4-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/temp-4-review\/","title":{"raw":"R1.5   Inequality Symbols and Graphs","rendered":"R1.5   Inequality Symbols and Graphs"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Represent inequalities using an inequality symbol<\/li>\r\n \t<li>Represent inequalities on a number line<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn inequality is a mathematical statement that compares two expressions using a phrase such as\u00a0<em>greater than<\/em> or <em>less than<\/em>. Special symbols are used in these statements. In algebra, inequalities are used to describe sets of values, as opposed to single values, of a variable. Sometimes, several numbers will satisfy an inequality, but at other times infinitely many numbers may provide solutions. Rather than try to list a possibly infinitely large set of numbers, mathematicians have developed some efficient ways to describe such large lists.\r\n<h2>Inequality Symbols<\/h2>\r\nOne way to represent such a list of numbers, an inequality, is by using an inequality symbol:\r\n<ul>\r\n \t<li>[latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than\u00a0[latex]9[\/latex]. Since this list is infinite, it would be impossilbe to list all numbers less than [latex]9[\/latex].<\/li>\r\n \t<li>[latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\r\n<\/ul>\r\nIf you were to read the above statement from left to right, it would translate as <em>[latex]-5[\/latex] is less than or equal to t<\/em>. The direction of the symbol is dependent upon the statement you wish to make. For example, the following statements are equivalent. Both represent the list of all numbers less than 9. Note how the open end of the inequality symbol faces the larger value while the smaller, pointy end points to the smaller of the values:\r\n<ul>\r\n \t<li>[latex]{x}\\lt{9}[\/latex]<\/li>\r\n \t<li>[latex]{9}\\gt{x}[\/latex]<\/li>\r\n<\/ul>\r\nHere's another way of looking at is:\r\n<ul>\r\n \t<li>[latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\r\n \t<li>[latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5 which is easier to interpret than 5 is less than x.<\/li>\r\n<\/ul>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign, as they would be translated reading from left to right.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Symbol<\/th>\r\n<th>Words<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\neq [\/latex]<\/td>\r\n<td>not equal to<\/td>\r\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<strong>is<\/strong> <b>not equal<\/b> to 8<em>.<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\gt[\/latex]<\/td>\r\n<td>greater than<\/td>\r\n<td>[latex]{5}\\gt{1}[\/latex], <i>5<\/i>\u00a0<strong>is greater than<\/strong>\u00a0<i>1<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\lt[\/latex]<\/td>\r\n<td>less than<\/td>\r\n<td>[latex]{2}\\lt{11}[\/latex], 2<i>\u00a0<\/i><b>is less than<\/b>\u00a0<i>11<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\geq [\/latex]<\/td>\r\n<td>greater than or equal to<\/td>\r\n<td>[latex]{4}\\geq{ 4}[\/latex], 4<i>\u00a0<\/i><b>is greater than or equal to<\/b>\u00a0<i>4<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\leq [\/latex]<\/td>\r\n<td>less than or equal to<\/td>\r\n<td>[latex]{7}\\leq{9}[\/latex], <i>7<\/i>\u00a0<b>is less than or equal to<\/b>\u00a0<i>9<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe inequality [latex]x&gt;y[\/latex]\u00a0can also be written as [latex]{y}&lt;{x}[\/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.\r\n<h2>Graphing an Inequality<\/h2>\r\nAnother way to represent an inequality is by graphing it on a number line:\r\n\r\n<img class=\"aligncenter wp-image-3855 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182808\/MITE_Lippman_Arithmetic_pdf__page_356_of_417_-300x58.png\" alt=\"A numberline. It is a long horizontal line with evenly spaced points, the middle of which is zero.\" width=\"300\" height=\"58\" \/>\r\n\r\nBelow are three examples of inequalities and their graphs. \u00a0Graphs are often helpful for visualizing information.\r\n\r\n[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to [latex]4[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image034.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 4. Shaded line through all numbers less than negative 4.\" width=\"575\" height=\"31\" \/>\r\n\r\n[latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image035.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 3. Shaded line through all numbers greater than negative 3.\" width=\"575\" height=\"31\" \/>\r\n\r\nEach of these graphs begins with a circle\u2014either an open or closed (shaded) circle. This point is often called the <i>end point<\/i> of the solution. A closed, or shaded, circle is used to represent the inequalities <i>greater than or equal to<\/i>\u00a0[latex] \\displaystyle \\left(\\geq\\right) [\/latex] or <i>less than or equal to<\/i>\u00a0[latex] \\displaystyle \\left(\\leq\\right) [\/latex]. The end point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The end point is <i>not <\/i>part of the solution. When the end point is not included in the solution, we often say we have <em>strict inequality<\/em> rather than <em>inequality with equality<\/em>.\r\n\r\nThe graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is <i>greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph the\u00a0inequality [latex]x\\ge 4[\/latex]\r\n[reveal-answer q=\"797241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797241\"]\r\n\r\nWe can use a number line as shown. Because the values for\u00a0[latex]x[\/latex] include\u00a0[latex]4[\/latex], we place a solid dot on the number line at\u00a0[latex]4[\/latex].\r\n\r\nThen we draw a line that\u00a0begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to\u00a0[latex]4[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182809\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis video shows an example of how to draw the graph of an inequality.\r\nhttps:\/\/youtu.be\/-kiAeGbSe5c\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite an inequality describing all the real numbers on the number line that are strictly less than\u00a0[latex]2[\/latex]. Then draw the corresponding graph.\r\n[reveal-answer q=\"867890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867890\"]\r\n\r\nWe need to start from the left and work right, so we start from negative infinity and end at [latex]2[\/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[\/latex].\r\n\r\nInequality: [latex]x\\lt2[\/latex]\r\n\r\nTo draw the graph, place an open dot on the number line first, and then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image033.jpg#fixme#fixme\" alt=\"Number line. Unshaded circle around 2 and shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Represent inequalities using an inequality symbol<\/li>\n<li>Represent inequalities on a number line<\/li>\n<\/ul>\n<\/div>\n<p>An inequality is a mathematical statement that compares two expressions using a phrase such as\u00a0<em>greater than<\/em> or <em>less than<\/em>. Special symbols are used in these statements. In algebra, inequalities are used to describe sets of values, as opposed to single values, of a variable. Sometimes, several numbers will satisfy an inequality, but at other times infinitely many numbers may provide solutions. Rather than try to list a possibly infinitely large set of numbers, mathematicians have developed some efficient ways to describe such large lists.<\/p>\n<h2>Inequality Symbols<\/h2>\n<p>One way to represent such a list of numbers, an inequality, is by using an inequality symbol:<\/p>\n<ul>\n<li>[latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than\u00a0[latex]9[\/latex]. Since this list is infinite, it would be impossilbe to list all numbers less than [latex]9[\/latex].<\/li>\n<li>[latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\n<\/ul>\n<p>If you were to read the above statement from left to right, it would translate as <em>[latex]-5[\/latex] is less than or equal to t<\/em>. The direction of the symbol is dependent upon the statement you wish to make. For example, the following statements are equivalent. Both represent the list of all numbers less than 9. Note how the open end of the inequality symbol faces the larger value while the smaller, pointy end points to the smaller of the values:<\/p>\n<ul>\n<li>[latex]{x}\\lt{9}[\/latex]<\/li>\n<li>[latex]{9}\\gt{x}[\/latex]<\/li>\n<\/ul>\n<p>Here&#8217;s another way of looking at is:<\/p>\n<ul>\n<li>[latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\n<li>[latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5 which is easier to interpret than 5 is less than x.<\/li>\n<\/ul>\n<p>The box below shows the symbol, meaning, and an example for each inequality sign, as they would be translated reading from left to right.<\/p>\n<table>\n<thead>\n<tr>\n<th>Symbol<\/th>\n<th>Words<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\neq[\/latex]<\/td>\n<td>not equal to<\/td>\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<strong>is<\/strong> <b>not equal<\/b> to 8<em>.<\/em><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\gt[\/latex]<\/td>\n<td>greater than<\/td>\n<td>[latex]{5}\\gt{1}[\/latex], <i>5<\/i>\u00a0<strong>is greater than<\/strong>\u00a0<i>1<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\lt[\/latex]<\/td>\n<td>less than<\/td>\n<td>[latex]{2}\\lt{11}[\/latex], 2<i>\u00a0<\/i><b>is less than<\/b>\u00a0<i>11<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\geq[\/latex]<\/td>\n<td>greater than or equal to<\/td>\n<td>[latex]{4}\\geq{ 4}[\/latex], 4<i>\u00a0<\/i><b>is greater than or equal to<\/b>\u00a0<i>4<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\leq[\/latex]<\/td>\n<td>less than or equal to<\/td>\n<td>[latex]{7}\\leq{9}[\/latex], <i>7<\/i>\u00a0<b>is less than or equal to<\/b>\u00a0<i>9<\/i><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The inequality [latex]x>y[\/latex]\u00a0can also be written as [latex]{y}<{x}[\/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.\n\n\n<h2>Graphing an Inequality<\/h2>\n<p>Another way to represent an inequality is by graphing it on a number line:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3855 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182808\/MITE_Lippman_Arithmetic_pdf__page_356_of_417_-300x58.png\" alt=\"A numberline. It is a long horizontal line with evenly spaced points, the middle of which is zero.\" width=\"300\" height=\"58\" \/><\/p>\n<p>Below are three examples of inequalities and their graphs. \u00a0Graphs are often helpful for visualizing information.<\/p>\n<p>[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to [latex]4[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image034.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 4. Shaded line through all numbers less than negative 4.\" width=\"575\" height=\"31\" \/><\/p>\n<p>[latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image035.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 3. Shaded line through all numbers greater than negative 3.\" width=\"575\" height=\"31\" \/><\/p>\n<p>Each of these graphs begins with a circle\u2014either an open or closed (shaded) circle. This point is often called the <i>end point<\/i> of the solution. A closed, or shaded, circle is used to represent the inequalities <i>greater than or equal to<\/i>\u00a0[latex]\\displaystyle \\left(\\geq\\right)[\/latex] or <i>less than or equal to<\/i>\u00a0[latex]\\displaystyle \\left(\\leq\\right)[\/latex]. The end point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The end point is <i>not <\/i>part of the solution. When the end point is not included in the solution, we often say we have <em>strict inequality<\/em> rather than <em>inequality with equality<\/em>.<\/p>\n<p>The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex]\\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is <i>greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph the\u00a0inequality [latex]x\\ge 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797241\">Show Solution<\/span><\/p>\n<div id=\"q797241\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use a number line as shown. Because the values for\u00a0[latex]x[\/latex] include\u00a0[latex]4[\/latex], we place a solid dot on the number line at\u00a0[latex]4[\/latex].<\/p>\n<p>Then we draw a line that\u00a0begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to\u00a0[latex]4[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182809\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This video shows an example of how to draw the graph of an inequality.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Graph Linear Inequalities in One Variable (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-kiAeGbSe5c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write an inequality describing all the real numbers on the number line that are strictly less than\u00a0[latex]2[\/latex]. Then draw the corresponding graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q867890\">Show Solution<\/span><\/p>\n<div id=\"q867890\" class=\"hidden-answer\" style=\"display: none\">\n<p>We need to start from the left and work right, so we start from negative infinity and end at [latex]2[\/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[\/latex].<\/p>\n<p>Inequality: [latex]x\\lt2[\/latex]<\/p>\n<p>To draw the graph, place an open dot on the number line first, and then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image033.jpg#fixme#fixme\" alt=\"Number line. Unshaded circle around 2 and shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/><\/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-96\">\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>Revision and Adaptation. <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>Solving One-Step Inequalities from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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>Graph Linear Inequalities in One Variable (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-kiAeGbSe5c\">https:\/\/youtu.be\/-kiAeGbSe5c<\/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":395986,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Solving One-Step Inequalities from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Graph Linear Inequalities in One Variable (Basic)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/-kiAeGbSe5c\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-96","chapter","type-chapter","status-publish","hentry"],"part":91,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/96","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/96\/revisions"}],"predecessor-version":[{"id":1485,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/96\/revisions\/1485"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/91"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/96\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=96"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=96"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=96"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=96"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}