{"id":1106,"date":"2024-05-06T19:02:01","date_gmt":"2024-05-06T19:02:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1106"},"modified":"2025-08-21T23:19:00","modified_gmt":"2025-08-21T23:19:00","slug":"linear-inequalities-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/linear-inequalities-learn-it-1\/","title":{"raw":"Linear Inequalities: Learn It 1","rendered":"Linear Inequalities: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Use interval notation to show solutions to inequalities.<\/li>\r\n \t<li>Solve linear inequalities by applying their properties.<\/li>\r\n<\/ul>\r\n<\/section>\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"212\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200407\/CNX_CAT_Figure_02_07_001N.jpg\" alt=\"Several red winner\u2019s ribbons lie on a white table.\" width=\"212\" height=\"141\" \/> Red award ribbons[\/caption]\r\n<h2 data-type=\"title\">Using Interval Notation<\/h2>\r\nIn many academic institutions, student performance is evaluated using a Grade Point Average (GPA) system where scores range from [latex]0[\/latex] to [latex]4[\/latex]. This scale can be represented using linear inequalities to define the complete range of possible GPAs. How could these GPA scores be expressed mathematically?\r\n<ul>\r\n \t<li><strong>Set-Builder Notation: <\/strong>[latex]\\{x \\mid 0 \\leq x \\leq 4\\}[\/latex], where [latex]x[\/latex] represents a student's GPA.\r\nThe inequality [latex]0 \\leq x \\leq 4[\/latex] encapsulates the idea that a GPA can be as low as [latex]0[\/latex], which might indicate failing grades or no coursework completed, and as high as [latex]4[\/latex], representing perfect academic performance.<\/li>\r\n \t<li><strong>Interval Notation: <\/strong>[latex][0, 4][\/latex].\r\nThis notation correctly specifies that the GPA can be any value including and between [latex]0[\/latex] and [latex]4[\/latex].<\/li>\r\n \t<li><strong>Number Line:<\/strong> The number line below illustrates that the solution set includes all real numbers between\u00a0[latex]0[\/latex] and [latex]4[\/latex].\r\n\r\n[caption id=\"attachment_1112\" align=\"aligncenter\" width=\"620\"]<img class=\"wp-image-1112 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM.png\" alt=\"\" width=\"620\" height=\"67\" \/> Number line with points on 0 and 4[\/caption]<\/li>\r\n<\/ul>\r\n<section class=\"textbox proTip\">Interval Notation:\r\n<ul>\r\n \t<li><strong>Parentheses<\/strong> represent solutions greater or less than the number.<\/li>\r\n \t<li><strong>Brackets<\/strong> represent solutions that are greater than or equal to or less than or equal to the number.<\/li>\r\n \t<li>Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \"equaled.\"<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>What is an Inequality?<\/strong>An inequality is a mathematical statement that compares two expressions using symbols such as:\r\n<ul>\r\n \t<li><strong>[latex]&lt;[\/latex]<\/strong> (less than)<\/li>\r\n \t<li><strong>[latex]&gt;[\/latex]<\/strong> (greater than)<\/li>\r\n \t<li><strong>[latex]\\leq[\/latex]<\/strong> (less than or equal to)<\/li>\r\n \t<li><strong>[latex]\\geq[\/latex]<\/strong> (greater than or equal to)<\/li>\r\n<\/ul>\r\nInequalities show that one side of the statement is not equal to the other. Instead, it describes a range of possible values. For example, the inequality [latex]x &gt; 3[\/latex] means that [latex]x[\/latex] can be any number greater than 3.\r\n\r\n<\/section>\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 18.2759%; text-align: center;\">Inequality<\/th>\r\n<th style=\"width: 25.351%; text-align: center;\">Words<\/th>\r\n<th style=\"width: 29.9891%; text-align: center;\">Interval Notation<\/th>\r\n<td style=\"width: 26.384%; text-align: center;\"><strong>Number Line<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1115\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1115 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194754\/Screenshot-2024-05-06-at-12.47.49%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with two open circles[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{x}\\gt{a}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1118\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1118 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194918\/Screenshot-2024-05-06-at-12.49.11%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with one open circle and a right facing arrow[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{x}\\lt{b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1119\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1119 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194957\/Screenshot-2024-05-06-at-12.49.53%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with one open circle and a left facing arrow[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{x}\\ge{a}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1121\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1121 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195059\/Screenshot-2024-05-06-at-12.50.55%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with one closed circle and a right facing arrow[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{x}\\le{b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1120\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1120 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195025\/Screenshot-2024-05-06-at-12.50.18%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with one closed circle and a left facing arrow[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1122\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1122 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195127\/Screenshot-2024-05-06-at-12.51.22%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with one closed circle and one open circle[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1123\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1123 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195154\/Screenshot-2024-05-06-at-12.51.48%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with one closed circle and one open circle[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1124\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1124 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195212\/Screenshot-2024-05-06-at-12.52.08%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with two closed circles[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1126\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1126 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195450\/Screenshot-2024-05-06-at-12.54.45%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with two open circles and arrows facing opposite directions[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 18.2759%;\">All real numbers<\/td>\r\n<td style=\"width: 25.351%;\">All real numbers<\/td>\r\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<td style=\"width: 26.384%;\">\r\n\r\n[caption id=\"attachment_1125\" align=\"aligncenter\" width=\"213\"]<img class=\"wp-image-1125 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195250\/Screenshot-2024-05-06-at-12.52.45%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" \/> Number line with two arrows[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Using interval notation to express the inequalities.\r\n<ol>\r\n \t<li>All real numbers greater than or equal to [latex]-2[\/latex].<\/li>\r\n \t<li>All real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].<\/li>\r\n \t<li>All real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].<\/li>\r\n \t<li>All real numbers less than [latex]-2[\/latex] or greater than or equal to [latex]3[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"249154\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"249154\"]\r\n<ol>\r\n \t<li>[latex]\\left[-2,\\infty \\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left[-3,5\\right][\/latex]<\/li>\r\n \t<li>[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex][\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Represent the number line below in words, inequality, and an interval notation.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"339\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182811\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"339\" height=\"95\" \/> Number line with open circle on 10 and left-facing arrow[\/caption]\r\n\r\n[reveal-answer q=\"354117\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"354117\"]\r\n<ul>\r\n \t<li>Words: all real numbers less than [latex]10[\/latex]<\/li>\r\n \t<li>Interval Notation: [latex](-\\infty, 10)[\/latex]<\/li>\r\n \t<li>Inequality: [latex]x&lt;10[\/latex][\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18942[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18943[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use interval notation to show solutions to inequalities.<\/li>\n<li>Solve linear inequalities by applying their properties.<\/li>\n<\/ul>\n<\/section>\n<figure style=\"width: 212px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200407\/CNX_CAT_Figure_02_07_001N.jpg\" alt=\"Several red winner\u2019s ribbons lie on a white table.\" width=\"212\" height=\"141\" \/><figcaption class=\"wp-caption-text\">Red award ribbons<\/figcaption><\/figure>\n<h2 data-type=\"title\">Using Interval Notation<\/h2>\n<p>In many academic institutions, student performance is evaluated using a Grade Point Average (GPA) system where scores range from [latex]0[\/latex] to [latex]4[\/latex]. This scale can be represented using linear inequalities to define the complete range of possible GPAs. How could these GPA scores be expressed mathematically?<\/p>\n<ul>\n<li><strong>Set-Builder Notation: <\/strong>[latex]\\{x \\mid 0 \\leq x \\leq 4\\}[\/latex], where [latex]x[\/latex] represents a student&#8217;s GPA.<br \/>\nThe inequality [latex]0 \\leq x \\leq 4[\/latex] encapsulates the idea that a GPA can be as low as [latex]0[\/latex], which might indicate failing grades or no coursework completed, and as high as [latex]4[\/latex], representing perfect academic performance.<\/li>\n<li><strong>Interval Notation: <\/strong>[latex][0, 4][\/latex].<br \/>\nThis notation correctly specifies that the GPA can be any value including and between [latex]0[\/latex] and [latex]4[\/latex].<\/li>\n<li><strong>Number Line:<\/strong> The number line below illustrates that the solution set includes all real numbers between\u00a0[latex]0[\/latex] and [latex]4[\/latex].<br \/>\n<figure id=\"attachment_1112\" aria-describedby=\"caption-attachment-1112\" style=\"width: 620px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1112 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM.png\" alt=\"\" width=\"620\" height=\"67\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM.png 620w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM-300x32.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM-65x7.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM-225x24.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194106\/Screenshot-2024-05-06-at-12.34.18%E2%80%AFPM-350x38.png 350w\" sizes=\"(max-width: 620px) 100vw, 620px\" \/><figcaption id=\"caption-attachment-1112\" class=\"wp-caption-text\">Number line with points on 0 and 4<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<section class=\"textbox proTip\">Interval Notation:<\/p>\n<ul>\n<li><strong>Parentheses<\/strong> represent solutions greater or less than the number.<\/li>\n<li><strong>Brackets<\/strong> represent solutions that are greater than or equal to or less than or equal to the number.<\/li>\n<li>Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be &#8220;equaled.&#8221;<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>What is an Inequality?<\/strong>An inequality is a mathematical statement that compares two expressions using symbols such as:<\/p>\n<ul>\n<li><strong>[latex]<[\/latex]<\/strong> (less than)<\/li>\n<li><strong>[latex]>[\/latex]<\/strong> (greater than)<\/li>\n<li><strong>[latex]\\leq[\/latex]<\/strong> (less than or equal to)<\/li>\n<li><strong>[latex]\\geq[\/latex]<\/strong> (greater than or equal to)<\/li>\n<\/ul>\n<p>Inequalities show that one side of the statement is not equal to the other. Instead, it describes a range of possible values. For example, the inequality [latex]x > 3[\/latex] means that [latex]x[\/latex] can be any number greater than 3.<\/p>\n<\/section>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 18.2759%; text-align: center;\">Inequality<\/th>\n<th style=\"width: 25.351%; text-align: center;\">Words<\/th>\n<th style=\"width: 29.9891%; text-align: center;\">Interval Notation<\/th>\n<td style=\"width: 26.384%; text-align: center;\"><strong>Number Line<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(a,b\\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1115\" aria-describedby=\"caption-attachment-1115\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1115 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194754\/Screenshot-2024-05-06-at-12.47.49%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194754\/Screenshot-2024-05-06-at-12.47.49%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194754\/Screenshot-2024-05-06-at-12.47.49%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1115\" class=\"wp-caption-text\">Number line with two open circles<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{x}\\gt{a}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1118\" aria-describedby=\"caption-attachment-1118\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1118 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194918\/Screenshot-2024-05-06-at-12.49.11%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194918\/Screenshot-2024-05-06-at-12.49.11%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194918\/Screenshot-2024-05-06-at-12.49.11%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1118\" class=\"wp-caption-text\">Number line with one open circle and a right facing arrow<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{x}\\lt{b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1119\" aria-describedby=\"caption-attachment-1119\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1119 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194957\/Screenshot-2024-05-06-at-12.49.53%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194957\/Screenshot-2024-05-06-at-12.49.53%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06194957\/Screenshot-2024-05-06-at-12.49.53%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1119\" class=\"wp-caption-text\">Number line with one open circle and a left facing arrow<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{x}\\ge{a}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1121\" aria-describedby=\"caption-attachment-1121\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1121 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195059\/Screenshot-2024-05-06-at-12.50.55%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195059\/Screenshot-2024-05-06-at-12.50.55%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195059\/Screenshot-2024-05-06-at-12.50.55%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1121\" class=\"wp-caption-text\">Number line with one closed circle and a right facing arrow<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{x}\\le{b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1120\" aria-describedby=\"caption-attachment-1120\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1120 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195025\/Screenshot-2024-05-06-at-12.50.18%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195025\/Screenshot-2024-05-06-at-12.50.18%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195025\/Screenshot-2024-05-06-at-12.50.18%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1120\" class=\"wp-caption-text\">Number line with one closed circle and a left facing arrow<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left[a,b\\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1122\" aria-describedby=\"caption-attachment-1122\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1122 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195127\/Screenshot-2024-05-06-at-12.51.22%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195127\/Screenshot-2024-05-06-at-12.51.22%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195127\/Screenshot-2024-05-06-at-12.51.22%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1122\" class=\"wp-caption-text\">Number line with one closed circle and one open circle<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(a,b\\right][\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1123\" aria-describedby=\"caption-attachment-1123\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1123 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195154\/Screenshot-2024-05-06-at-12.51.48%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195154\/Screenshot-2024-05-06-at-12.51.48%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195154\/Screenshot-2024-05-06-at-12.51.48%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1123\" class=\"wp-caption-text\">Number line with one closed circle and one open circle<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left[a,b\\right][\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1124\" aria-describedby=\"caption-attachment-1124\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1124 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195212\/Screenshot-2024-05-06-at-12.52.08%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195212\/Screenshot-2024-05-06-at-12.52.08%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195212\/Screenshot-2024-05-06-at-12.52.08%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1124\" class=\"wp-caption-text\">Number line with two closed circles<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\n<td style=\"width: 25.351%;\">All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1126\" aria-describedby=\"caption-attachment-1126\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1126 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195450\/Screenshot-2024-05-06-at-12.54.45%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195450\/Screenshot-2024-05-06-at-12.54.45%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195450\/Screenshot-2024-05-06-at-12.54.45%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1126\" class=\"wp-caption-text\">Number line with two open circles and arrows facing opposite directions<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 18.2759%;\">All real numbers<\/td>\n<td style=\"width: 25.351%;\">All real numbers<\/td>\n<td style=\"width: 29.9891%; text-align: center;\">[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<td style=\"width: 26.384%;\">\n<figure id=\"attachment_1125\" aria-describedby=\"caption-attachment-1125\" style=\"width: 213px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1125 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195250\/Screenshot-2024-05-06-at-12.52.45%E2%80%AFPM.png\" alt=\"\" width=\"213\" height=\"46\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195250\/Screenshot-2024-05-06-at-12.52.45%E2%80%AFPM.png 213w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/06195250\/Screenshot-2024-05-06-at-12.52.45%E2%80%AFPM-65x14.png 65w\" sizes=\"(max-width: 213px) 100vw, 213px\" \/><figcaption id=\"caption-attachment-1125\" class=\"wp-caption-text\">Number line with two arrows<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Using interval notation to express the inequalities.<\/p>\n<ol>\n<li>All real numbers greater than or equal to [latex]-2[\/latex].<\/li>\n<li>All real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].<\/li>\n<li>All real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].<\/li>\n<li>All real numbers less than [latex]-2[\/latex] or greater than or equal to [latex]3[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q249154\">Show Answer<\/button><\/p>\n<div id=\"q249154\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\left[-2,\\infty \\right)[\/latex]<\/li>\n<li>[latex]\\left[-3,5\\right][\/latex]<\/li>\n<li>[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/li>\n<li>[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Represent the number line below in words, inequality, and an interval notation.<\/p>\n<figure style=\"width: 339px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182811\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"339\" height=\"95\" \/><figcaption class=\"wp-caption-text\">Number line with open circle on 10 and left-facing arrow<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q354117\">Show Answer<\/button><\/p>\n<div id=\"q354117\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Words: all real numbers less than [latex]10[\/latex]<\/li>\n<li>Interval Notation: [latex](-\\infty, 10)[\/latex]<\/li>\n<li>Inequality: [latex]x<10[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18942\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18942&theme=lumen&iframe_resize_id=ohm18942&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18943\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18943&theme=lumen&iframe_resize_id=ohm18943&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":75,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1106"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":30,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1106\/revisions"}],"predecessor-version":[{"id":7977,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1106\/revisions\/7977"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/75"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1106\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=1106"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1106"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=1106"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=1106"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}