{"id":1058,"date":"2021-08-18T14:57:39","date_gmt":"2021-08-18T14:57:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1058"},"modified":"2022-01-26T23:04:19","modified_gmt":"2022-01-26T23:04:19","slug":"inequality-symbols-and-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/inequality-symbols-and-graphs\/","title":{"raw":"Inequality Symbols and Graphs","rendered":"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 style=\"font-weight: 400;\" aria-level=\"1\">Represent inequalities using an inequality symbol<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Represent inequalities on a number line<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn inequality compares two expressions, identifying one as more, less, or simply different than the other. We will review the mathematical symbols used to represent various relationships between two quantities. Values of a variable which make a statement true are called the solution set. There may be a few or infinitely many numbers which satisfy an inequality. We will also review ways to represent these solution sets.\r\n\r\nThe symbol \u201c&lt;\u201d means \u201cless than.\u201d\u00a0 For example, 3 &lt; 5.\u00a0 The symbol \u201c&gt;\u201d means \u201cgreater than.\u201d\u00a0 You can think of the inequality as an arrow which points to the smaller number. Or, you may have learned to think of the inequality as a hungry fish\u2019s mouth, about to eat the larger number.\r\n<h2>Inequality Symbols<\/h2>\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\r\nFor example, [latex]2&lt;7[\/latex] is true, as is [latex]7&gt;2[\/latex].\u00a0 So the statement [latex]2&lt;x[\/latex] can also be written [latex]x&gt;2[\/latex].\u00a0 Any real number larger than [latex]2[\/latex] is a solution to the inequality.\r\n\r\nThe expressions on each side of an inequality can be interchanged 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 <strong>real number line<\/strong>:\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\nConsider the inequality [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\nConsider the inequality [latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to [latex]-3[\/latex].\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>\r\n<h4>Note<\/h4>\r\nSometimes we are only interested in integer values that satisfy an inequality.\u00a0 Suppose [latex]x[\/latex] represents the number of cars a person owns and we are told [latex]x &lt; 2[\/latex].\u00a0 Since no one can own a negative number of cars, our solutions set would be the nonnegative integers which are less than [latex]2[\/latex].\u00a0 Then our possible solutions are [latex]0[\/latex] and [latex]1[\/latex].\u00a0 We would graph these individual points by plotting solid dots at these values.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Represent inequalities using an inequality symbol<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Represent inequalities on a number line<\/li>\n<\/ul>\n<\/div>\n<p>An inequality compares two expressions, identifying one as more, less, or simply different than the other. We will review the mathematical symbols used to represent various relationships between two quantities. Values of a variable which make a statement true are called the solution set. There may be a few or infinitely many numbers which satisfy an inequality. We will also review ways to represent these solution sets.<\/p>\n<p>The symbol \u201c&lt;\u201d means \u201cless than.\u201d\u00a0 For example, 3 &lt; 5.\u00a0 The symbol \u201c&gt;\u201d means \u201cgreater than.\u201d\u00a0 You can think of the inequality as an arrow which points to the smaller number. Or, you may have learned to think of the inequality as a hungry fish\u2019s mouth, about to eat the larger number.<\/p>\n<h2>Inequality Symbols<\/h2>\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\nFor example, [latex]2<7[\/latex] is true, as is [latex]7>2[\/latex].\u00a0 So the statement [latex]2<x[\/latex] can also be written [latex]x>2[\/latex].\u00a0 Any real number larger than [latex]2[\/latex] is a solution to the inequality.<\/p>\n<p>The expressions on each side of an inequality can be interchanged as long as the inequality symbol between them is also reversed.<\/p>\n<h2>Graphing an Inequality<\/h2>\n<p>Another way to represent an inequality is by graphing it on a <strong>real number line<\/strong>:<\/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>Consider the inequality [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>Consider the inequality [latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to [latex]-3[\/latex].<\/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<h4>Note<\/h4>\n<p>Sometimes we are only interested in integer values that satisfy an inequality.\u00a0 Suppose [latex]x[\/latex] represents the number of cars a person owns and we are told [latex]x < 2[\/latex].\u00a0 Since no one can own a negative number of cars, our solutions set would be the nonnegative integers which are less than [latex]2[\/latex].\u00a0 Then our possible solutions are [latex]0[\/latex] and [latex]1[\/latex].\u00a0 We would graph these individual points by plotting solid dots at these values.\n<\/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-1058\">\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>: Graph Linear Inequalities in One Variable (Basic). <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":169134,"menu_order":3,"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 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