{"id":1397,"date":"2024-05-10T23:29:12","date_gmt":"2024-05-10T23:29:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1397"},"modified":"2025-08-21T23:27:47","modified_gmt":"2025-08-21T23:27:47","slug":"module-5-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-5-background-youll-need-1\/","title":{"raw":"Function Basic: Background You'll Need 1","rendered":"Function Basic: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use set-builder, inequality, and interval notations to describe sets of numbers on a number line<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"font-600 text-xl font-bold\">Set Notation: Inequality, Set-Builder, and Interval Notations<\/h2>\r\nIn mathematics, we often need to describe sets of numbers that satisfy certain conditions. There are several ways to represent these sets, each with its own advantages and uses in different contexts. This page introduces three common notations for describing sets of numbers: inequality notation, set-builder notation, and interval notation.\r\n\r\n<strong>Inequality notation<\/strong> uses the symbols [latex]&lt;[\/latex], [latex]&gt;[\/latex], [latex]\\le[\/latex], and [latex]\\ge[\/latex] to describe ranges of numbers.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x &gt; 3[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]2 \\le x &lt; 5[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><strong>Set-builder notation<\/strong> uses curly braces [latex]{}[\/latex] and a vertical bar [latex]|[\/latex] to describe sets based on their properties.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{x \\mid x &gt; 3}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than [latex]3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{x \\mid 2 \\le x &lt; 5}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><strong>Interval notation<\/strong> uses parentheses [latex]([\/latex] [latex])[\/latex] and square brackets [latex][[\/latex] [latex]][\/latex] to represent continuous ranges of numbers.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex](3, \\infty)[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex][2, 5)[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>The table below compares inequality notation, set-builder notation, and interval notation.\r\n<table style=\"height: 312px;\">\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px; width: 194px;\"><\/th>\r\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\r\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\r\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\">\r\n\r\n[caption id=\"attachment_12492\" align=\"aligncenter\" width=\"265\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img class=\"wp-image-12492 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a> Number line examining circles at 5 and 10[\/caption]<\/td>\r\n<td style=\"height: 45px; width: 148px;\">[latex]5 \\lt h \\le 10[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 &lt; h \\le 10\\}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 48px;\">\r\n<td style=\"height: 48px; width: 194px;\">\r\n\r\n[caption id=\"attachment_12493\" align=\"aligncenter\" width=\"281\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img class=\"wp-image-12493 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a> Number line examining circles at 5 and 10[\/caption]<\/td>\r\n<td style=\"height: 48px; width: 148px;\">[latex]5 \\le\u00a0 h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 49px;\">\r\n<td style=\"height: 49px; width: 194px;\">\r\n\r\n[caption id=\"attachment_12494\" align=\"aligncenter\" width=\"283\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img class=\"wp-image-12494 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a> Number line examining circles at 5 and 10[\/caption]<\/td>\r\n<td style=\"height: 49px; width: 148px;\">[latex]5 \\lt h\\lt 10[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 &lt; h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 51px;\">\r\n<td style=\"height: 51px; width: 194px;\">\r\n\r\n[caption id=\"attachment_12495\" align=\"aligncenter\" width=\"271\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img class=\"wp-image-12495 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a> Number line examining circles or arrows at 5 and 10[\/caption]<\/td>\r\n<td style=\"height: 51px; width: 148px;\">[latex]h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"height: 44px; width: 194px;\">\r\n\r\n[caption id=\"attachment_12496\" align=\"aligncenter\" width=\"310\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img class=\"wp-image-12496 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a> Number line examining circles and arrows at 10[\/caption]<\/td>\r\n<td style=\"height: 44px; width: 148px;\">[latex]h&gt;10[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h &gt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\">\r\n\r\n[caption id=\"attachment_12497\" align=\"aligncenter\" width=\"359\"]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img class=\"wp-image-12497 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a> Number line examining arrows at 5 and 10[\/caption]<\/td>\r\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"font-600 text-xl font-bold\"><strong>Special Cases and Symbols<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\infty[\/latex] (infinity) is used in interval notation to represent unbounded intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The empty set is represented as [latex]\\emptyset[\/latex] or [latex]\\{\\}[\/latex] in set-builder notation, and as [latex][\u00a0 ][\/latex] in interval notation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The union of sets is represented by the symbol [latex]\\cup[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/> Number line[\/caption]\r\n\r\nTo describe the values, [latex]x[\/latex], included in the intervals shown, we would say, \" [latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\"\r\n<table style=\"width: 102.547%;\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 19.1257%;\"><strong>Inequality<\/strong><\/td>\r\n<td style=\"width: 82.1494%;\">[latex]1\\le x\\le 3\\text{ or }x&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.1257%;\"><strong>Set-builder notation<\/strong><\/td>\r\n<td style=\"width: 82.1494%;\">[latex]\\left\\{x|1\\le x\\le 3 \\text{ or }x&gt;5\\right\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.1257%;\"><strong>Interval notation<\/strong><\/td>\r\n<td style=\"width: 82.1494%;\">[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]19119[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]19121[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]19122[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use set-builder, inequality, and interval notations to describe sets of numbers on a number line<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"font-600 text-xl font-bold\">Set Notation: Inequality, Set-Builder, and Interval Notations<\/h2>\n<p>In mathematics, we often need to describe sets of numbers that satisfy certain conditions. There are several ways to represent these sets, each with its own advantages and uses in different contexts. This page introduces three common notations for describing sets of numbers: inequality notation, set-builder notation, and interval notation.<\/p>\n<p><strong>Inequality notation<\/strong> uses the symbols [latex]<[\/latex], [latex]>[\/latex], [latex]\\le[\/latex], and [latex]\\ge[\/latex] to describe ranges of numbers.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]x > 3[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]2 \\le x < 5[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p><strong>Set-builder notation<\/strong> uses curly braces [latex]{}[\/latex] and a vertical bar [latex]|[\/latex] to describe sets based on their properties.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]{x \\mid x > 3}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{x \\mid 2 \\le x < 5}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p><strong>Interval notation<\/strong> uses parentheses [latex]([\/latex] [latex])[\/latex] and square brackets [latex][[\/latex] [latex]][\/latex] to represent continuous ranges of numbers.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex](3, \\infty)[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex][2, 5)[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p>The table below compares inequality notation, set-builder notation, and interval notation.<\/p>\n<table style=\"height: 312px;\">\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px; width: 194px;\"><\/th>\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\">\n<figure id=\"attachment_12492\" aria-describedby=\"caption-attachment-12492\" style=\"width: 265px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12492 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><figcaption id=\"caption-attachment-12492\" class=\"wp-caption-text\">Number line examining circles at 5 and 10<\/figcaption><\/figure>\n<\/td>\n<td style=\"height: 45px; width: 148px;\">[latex]5 \\lt h \\le 10[\/latex]<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 < h \\le 10\\}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 194px;\">\n<figure id=\"attachment_12493\" aria-describedby=\"caption-attachment-12493\" style=\"width: 281px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12493 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><figcaption id=\"caption-attachment-12493\" class=\"wp-caption-text\">Number line examining circles at 5 and 10<\/figcaption><\/figure>\n<\/td>\n<td style=\"height: 48px; width: 148px;\">[latex]5 \\le\u00a0 h<10[\/latex]<\/td>\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h < 10\\}[\/latex]<\/td>\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 49px;\">\n<td style=\"height: 49px; width: 194px;\">\n<figure id=\"attachment_12494\" aria-describedby=\"caption-attachment-12494\" style=\"width: 283px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12494 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><figcaption id=\"caption-attachment-12494\" class=\"wp-caption-text\">Number line examining circles at 5 and 10<\/figcaption><\/figure>\n<\/td>\n<td style=\"height: 49px; width: 148px;\">[latex]5 \\lt h\\lt 10[\/latex]<\/td>\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 < h < 10\\}[\/latex]<\/td>\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 51px;\">\n<td style=\"height: 51px; width: 194px;\">\n<figure id=\"attachment_12495\" aria-describedby=\"caption-attachment-12495\" style=\"width: 271px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12495 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><figcaption id=\"caption-attachment-12495\" class=\"wp-caption-text\">Number line examining circles or arrows at 5 and 10<\/figcaption><\/figure>\n<\/td>\n<td style=\"height: 51px; width: 148px;\">[latex]h<10[\/latex]<\/td>\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h < 10\\}[\/latex]<\/td>\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px; width: 194px;\">\n<figure id=\"attachment_12496\" aria-describedby=\"caption-attachment-12496\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12496 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><figcaption id=\"caption-attachment-12496\" class=\"wp-caption-text\">Number line examining circles and arrows at 10<\/figcaption><\/figure>\n<\/td>\n<td style=\"height: 44px; width: 148px;\">[latex]h>10[\/latex]<\/td>\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h > 10\\}[\/latex]<\/td>\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\">\n<figure id=\"attachment_12497\" aria-describedby=\"caption-attachment-12497\" style=\"width: 359px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-12497 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><figcaption id=\"caption-attachment-12497\" class=\"wp-caption-text\">Number line examining arrows at 5 and 10<\/figcaption><\/figure>\n<\/td>\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"font-600 text-xl font-bold\"><strong>Special Cases and Symbols<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\infty[\/latex] (infinity) is used in interval notation to represent unbounded intervals<\/li>\n<li class=\"whitespace-normal break-words\">The empty set is represented as [latex]\\emptyset[\/latex] or [latex]\\{\\}[\/latex] in set-builder notation, and as [latex][\u00a0 ][\/latex] in interval notation<\/li>\n<li class=\"whitespace-normal break-words\">The union of sets is represented by the symbol [latex]\\cup[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/><figcaption class=\"wp-caption-text\">Number line<\/figcaption><\/figure>\n<p>To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, &#8221; [latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.&#8221;<\/p>\n<table style=\"width: 102.547%;\" summary=\"..\">\n<tbody>\n<tr>\n<td style=\"width: 19.1257%;\"><strong>Inequality<\/strong><\/td>\n<td style=\"width: 82.1494%;\">[latex]1\\le x\\le 3\\text{ or }x>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.1257%;\"><strong>Set-builder notation<\/strong><\/td>\n<td style=\"width: 82.1494%;\">[latex]\\left\\{x|1\\le x\\le 3 \\text{ or }x>5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.1257%;\"><strong>Interval notation<\/strong><\/td>\n<td style=\"width: 82.1494%;\">[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. 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