{"id":1190,"date":"2024-05-06T23:12:13","date_gmt":"2024-05-06T23:12:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1190"},"modified":"2025-08-21T23:19:59","modified_gmt":"2025-08-21T23:19:59","slug":"linear-inequalities-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/linear-inequalities-fresh-take\/","title":{"raw":"Linear Inequalities: Fresh Take","rendered":"Linear Inequalities: Fresh Take"},"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<h2 data-type=\"title\">Using Interval Notation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Interval Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A concise way to represent ranges of real numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Incorporates infinity symbols for unbounded intervals<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inequalities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Mathematical statements comparing two expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use symbols like &lt;, &gt;, \u2264, \u2265 to show relationships between values<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set-Builder Notation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Another way to represent sets of numbers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex]{x \\mid 0 \\leq x \\leq 4}[\/latex] represents all [latex]x[\/latex] between [latex]0[\/latex] and [latex]4[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Number Line Representation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Visual method to display intervals and inequalities<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Uses solid dots for inclusive endpoints, open circles for exclusive endpoints<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Points<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Interval Types:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Open interval: [latex](a, b)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Closed interval: [latex][a, b][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Half-open intervals: [latex][a, b)[\/latex] or [latex](a, b][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Infinite intervals: [latex](a, \\infty)[\/latex] or [latex](-\\infty, b)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Union of Intervals:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Represented by the \u222a symbol<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Example: [latex](-\\infty, a) \\cup (b, \\infty)[\/latex] represents all numbers less than [latex]a[\/latex] or greater than [latex]b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Translating Between Notations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Inequality \u2192 Interval Notation \u2192 Number Line \u2192 Words<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each representation conveys the same information in a different format<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Describe the inequality [latex]x\\ge 4[\/latex] using interval notation\r\n[reveal-answer q=\"440342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"440342\"]The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].Note the use of a bracket on the left because 4 is included in the solution set.[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Express the following in interval notation, as an inequality, and on a number line:\r\n<p style=\"text-align: center;\">\"All real numbers greater than [latex]-2[\/latex] but less than or equal to [latex]5[\/latex]\"<\/p>\r\n[reveal-answer q=\"658591\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"658591\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Interval Notation: [latex](-2, 5][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inequality: [latex]-2 &lt; x \\leq 5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Number Line:\r\n\r\n[caption id=\"attachment_3489\" align=\"alignnone\" width=\"880\"]<img class=\"wp-image-3489 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855.png\" alt=\"A number line with a bold blue segment. The segment starts from an open circle at -2 and ends at a closed circle at 5. This representation indicates that the interval is ( \u2212 2 , 5 ] (\u22122,5], which includes all numbers between -2 and 5, where -2 is not included, but 5 is included.\" width=\"880\" height=\"381\" \/> Number line with open circle at -2 and closed circle at 5[\/caption]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Using the Properties of Inequalities<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Addition Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]a &lt; b[\/latex], then [latex]a + c &lt; b + c[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adding or subtracting the same number on both sides preserves the inequality<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplication Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]a &lt; b[\/latex] and [latex]c &gt; 0[\/latex], then [latex]ac &lt; bc[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]a &lt; b[\/latex] and [latex]c &lt; 0[\/latex], then [latex]ac &gt; bc[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplying or dividing by a positive number preserves the inequality<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiplying or dividing by a negative number reverses the inequality<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These properties apply to [latex]\\leq[\/latex], [latex]&gt;[\/latex], and [latex]\\ge[\/latex] as well<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Points<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solving Inequalities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Similar to solving equations, but with attention to inequality direction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate the variable on one side of the inequality<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perform the same operations on both sides<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reversing Inequalities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">When multiplying or dividing by a negative number, reverse the inequality sign<\/li>\r\n \t<li class=\"whitespace-normal break-words\">This is crucial for maintaining the correct solution set<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution Sets:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Often expressed in interval notation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Represent all real numbers that satisfy the inequality<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\">We are going to look at a line with endpoints along the x-axis.\r\n<ol>\r\n \t<li>First we will adjust the left endpoint to (-15,0), and the right endpoint to (5,0)\r\n\r\n[caption id=\"attachment_6714\" align=\"alignnone\" width=\"473\"]<img class=\"wp-image-6714\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08183340\/Screen-Shot-2019-07-08-at-11.26.00-AM.png\" alt=\"A line with endpoints at (-15,0) and (5,0).\" width=\"473\" height=\"161\" \/> A line with endpoints at (-15,0) and (5,0)[\/caption]<\/li>\r\n \t<li>Write an inequality that represents the line you created.<\/li>\r\n<\/ol>\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n3. If we were to slide the left endpoint to (2,0), what do you think will happen to the line?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n4. Now what if we were to slide the right endpoint to (11,0), what do you think will happen to the line? Sketch on a piece of paper what you think this new inequality graph will look like.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"748650\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"748650\"]\r\n\r\nWith endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write [latex]-15&lt;x&lt;5[\/latex]. We made it a strict inequality because the dots on the endpoints of the lines are open.\r\n\r\nMoving the left endpoint towards the right endpoint shortens the line. Then moving the right endpoint away from the left endpoint lengthens the line again.\r\n\r\n[caption id=\"attachment_6715\" align=\"alignnone\" width=\"317\"]<img class=\"wp-image-6715\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08183932\/Screen-Shot-2019-07-08-at-11.26.35-AM.png\" alt=\"Line with endpoints at (2,0) and (11,0).\" width=\"317\" height=\"128\" \/> Line with endpoints at (2,0) and (11,0)[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeagdcfe-X0xrHKgbDT0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/X0xrHKgbDT0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aeagdcfe-X0xrHKgbDT0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844233&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aeagdcfe-X0xrHKgbDT0&vembed=0&video_id=X0xrHKgbDT0&video_target=tpm-plugin-aeagdcfe-X0xrHKgbDT0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Graph+Basic+Inequalities+and+Express+Using+Interval+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Graph Basic Inequalities and Express Using Interval Notation\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3x - 2&lt;1[\/latex].[reveal-answer q=\"68318\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"68318\"][latex]x&lt;1[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve [latex]4x+7\\ge 2x - 3[\/latex].[reveal-answer q=\"32307\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"32307\"][latex]x\\ge -5[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following two videos for a demonstration of using the addition and multiplication properties to solve inequalities.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cfecedfg-1Z22Xh66VFM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1Z22Xh66VFM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cfecedfg-1Z22Xh66VFM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844234&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cfecedfg-1Z22Xh66VFM&vembed=0&video_id=1Z22Xh66VFM&video_target=tpm-plugin-cfecedfg-1Z22Xh66VFM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Solving+One+Step+Inequalities+by+Adding+and+Subtracting+(Variable+Left+Side)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the inequality and write the answer using interval notation: [latex]-x+4&lt;\\frac{1}{2}x+1[\/latex].[reveal-answer q=\"703883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703883\"][latex]\\left(2,\\infty \\right)[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\">Solve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].[reveal-answer q=\"524889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"524889\"][latex]\\left[-\\frac{3}{14},\\infty \\right)[\/latex][\/hidden-answer]<\/section>\r\n<h2>Compound Inequalities<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A compound inequality combines two inequalities in one statement<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can be connected by \"and\" (conjunction) or \"or\" (disjunction)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Compound Inequalities:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">\"And\" Compound Inequality: Both conditions must be true (e.g., [latex]a &lt; x &lt; b[\/latex])\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Represent a range of values between two bounds<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution is the intersection of the two individual inequalities<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">\"Or\" Compound Inequality: At least one condition must be true (e.g., [latex]x &lt; a \\text{ or } x &gt; b[\/latex])\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Represent values satisfying at least one of the conditions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solution is the union of the two individual inequalities<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Methods:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Separate into two inequalities and solve individually<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the compound inequality intact, operating on all parts simultaneously<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the same operation to all parts of the inequality<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pay attention to inequality direction when multiplying or dividing by negatives<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine results for the final solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the compound inequality [latex]4 &lt; 2x - 8\\le 10[\/latex].[reveal-answer q=\"265531\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265531\"][latex]6 &lt; x\\le 9\\text{ }\\text{ }\\text{or}\\left(6,9\\right][\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Solve the compound inequality: [latex]3y &lt; 4 - 5y &lt; 5+3y[\/latex].[reveal-answer q=\"661493\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"661493\"][latex]\\left(-\\frac{1}{8},\\frac{1}{2}\\right)[\/latex][\/hidden-answer]<\/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<h2 data-type=\"title\">Using Interval Notation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Interval Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A concise way to represent ranges of real numbers<\/li>\n<li class=\"whitespace-normal break-words\">Uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints<\/li>\n<li class=\"whitespace-normal break-words\">Incorporates infinity symbols for unbounded intervals<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inequalities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Mathematical statements comparing two expressions<\/li>\n<li class=\"whitespace-normal break-words\">Use symbols like &lt;, &gt;, \u2264, \u2265 to show relationships between values<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Set-Builder Notation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Another way to represent sets of numbers<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex]{x \\mid 0 \\leq x \\leq 4}[\/latex] represents all [latex]x[\/latex] between [latex]0[\/latex] and [latex]4[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Number Line Representation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Visual method to display intervals and inequalities<\/li>\n<li class=\"whitespace-normal break-words\">Uses solid dots for inclusive endpoints, open circles for exclusive endpoints<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Points<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Interval Types:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Open interval: [latex](a, b)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Closed interval: [latex][a, b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Half-open intervals: [latex][a, b)[\/latex] or [latex](a, b][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Infinite intervals: [latex](a, \\infty)[\/latex] or [latex](-\\infty, b)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Union of Intervals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represented by the \u222a symbol<\/li>\n<li class=\"whitespace-normal break-words\">Example: [latex](-\\infty, a) \\cup (b, \\infty)[\/latex] represents all numbers less than [latex]a[\/latex] or greater than [latex]b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Translating Between Notations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Inequality \u2192 Interval Notation \u2192 Number Line \u2192 Words<\/li>\n<li class=\"whitespace-normal break-words\">Each representation conveys the same information in a different format<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Describe the inequality [latex]x\\ge 4[\/latex] using interval notation<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q440342\">Show Solution<\/button><\/p>\n<div id=\"q440342\" class=\"hidden-answer\" style=\"display: none\">The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].Note the use of a bracket on the left because 4 is included in the solution set.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Express the following in interval notation, as an inequality, and on a number line:<\/p>\n<p style=\"text-align: center;\">&#8220;All real numbers greater than [latex]-2[\/latex] but less than or equal to [latex]5[\/latex]&#8220;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q658591\">Show Answer<\/button><\/p>\n<div id=\"q658591\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Interval Notation: [latex](-2, 5][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inequality: [latex]-2 < x \\leq 5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Number Line:<br \/>\n<figure id=\"attachment_3489\" aria-describedby=\"caption-attachment-3489\" style=\"width: 880px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3489 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855.png\" alt=\"A number line with a bold blue segment. The segment starts from an open circle at -2 and ends at a closed circle at 5. This representation indicates that the interval is ( \u2212 2 , 5 &#093; (\u22122,5&#093;, which includes all numbers between -2 and 5, where -2 is not included, but 5 is included.\" width=\"880\" height=\"381\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855.png 880w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855-300x130.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855-768x333.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855-65x28.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855-225x97.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/04163926\/Screenshot-2024-09-04-123855-350x152.png 350w\" sizes=\"(max-width: 880px) 100vw, 880px\" \/><figcaption id=\"caption-attachment-3489\" class=\"wp-caption-text\">Number line with open circle at -2 and closed circle at 5<\/figcaption><\/figure>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<h2>Using the Properties of Inequalities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Addition Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]a < b[\/latex], then [latex]a + c < b + c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Adding or subtracting the same number on both sides preserves the inequality<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiplication Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]a < b[\/latex] and [latex]c > 0[\/latex], then [latex]ac < bc[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]a < b[\/latex] and [latex]c < 0[\/latex], then [latex]ac > bc[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Multiplying or dividing by a positive number preserves the inequality<\/li>\n<li class=\"whitespace-normal break-words\">Multiplying or dividing by a negative number reverses the inequality<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">These properties apply to [latex]\\leq[\/latex], [latex]>[\/latex], and [latex]\\ge[\/latex] as well<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Points<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solving Inequalities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Similar to solving equations, but with attention to inequality direction<\/li>\n<li class=\"whitespace-normal break-words\">Isolate the variable on one side of the inequality<\/li>\n<li class=\"whitespace-normal break-words\">Perform the same operations on both sides<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reversing Inequalities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">When multiplying or dividing by a negative number, reverse the inequality sign<\/li>\n<li class=\"whitespace-normal break-words\">This is crucial for maintaining the correct solution set<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solution Sets:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often expressed in interval notation<\/li>\n<li class=\"whitespace-normal break-words\">Represent all real numbers that satisfy the inequality<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\">We are going to look at a line with endpoints along the x-axis.<\/p>\n<ol>\n<li>First we will adjust the left endpoint to (-15,0), and the right endpoint to (5,0)<br \/>\n<figure id=\"attachment_6714\" aria-describedby=\"caption-attachment-6714\" style=\"width: 473px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6714\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08183340\/Screen-Shot-2019-07-08-at-11.26.00-AM.png\" alt=\"A line with endpoints at (-15,0) and (5,0).\" width=\"473\" height=\"161\" \/><figcaption id=\"caption-attachment-6714\" class=\"wp-caption-text\">A line with endpoints at (-15,0) and (5,0)<\/figcaption><\/figure>\n<\/li>\n<li>Write an inequality that represents the line you created.<\/li>\n<\/ol>\n<p><textarea rows=\"1\"><\/textarea><\/p>\n<p>3. If we were to slide the left endpoint to (2,0), what do you think will happen to the line?<\/p>\n<p><textarea rows=\"1\"><\/textarea><\/p>\n<p>4. Now what if we were to slide the right endpoint to (11,0), what do you think will happen to the line? Sketch on a piece of paper what you think this new inequality graph will look like.<\/p>\n<p><textarea rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q748650\">Show Solution<\/button><\/p>\n<div id=\"q748650\" class=\"hidden-answer\" style=\"display: none\">\n<p>With endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write [latex]-15<x<5[\/latex]. We made it a strict inequality because the dots on the endpoints of the lines are open.\n\nMoving the left endpoint towards the right endpoint shortens the line. Then moving the right endpoint away from the left endpoint lengthens the line again.\n\n\n\n<figure id=\"attachment_6715\" aria-describedby=\"caption-attachment-6715\" style=\"width: 317px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6715\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08183932\/Screen-Shot-2019-07-08-at-11.26.35-AM.png\" alt=\"Line with endpoints at (2,0) and (11,0).\" width=\"317\" height=\"128\" \/><figcaption id=\"caption-attachment-6715\" class=\"wp-caption-text\">Line with endpoints at (2,0) and (11,0)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aeagdcfe-X0xrHKgbDT0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/X0xrHKgbDT0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aeagdcfe-X0xrHKgbDT0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844233&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aeagdcfe-X0xrHKgbDT0&#38;vembed=0&#38;video_id=X0xrHKgbDT0&#38;video_target=tpm-plugin-aeagdcfe-X0xrHKgbDT0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Graph+Basic+Inequalities+and+Express+Using+Interval+Notation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Graph Basic Inequalities and Express Using Interval Notation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]3x - 2<1[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q68318\">Show Solution<\/button><\/p>\n<div id=\"q68318\" class=\"hidden-answer\" style=\"display: none\">[latex]x<1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve [latex]4x+7\\ge 2x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q32307\">Show Solution<\/button><\/p>\n<div id=\"q32307\" class=\"hidden-answer\" style=\"display: none\">[latex]x\\ge -5[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following two videos for a demonstration of using the addition and multiplication properties to solve inequalities.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cfecedfg-1Z22Xh66VFM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/1Z22Xh66VFM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cfecedfg-1Z22Xh66VFM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844234&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cfecedfg-1Z22Xh66VFM&#38;vembed=0&#38;video_id=1Z22Xh66VFM&#38;video_target=tpm-plugin-cfecedfg-1Z22Xh66VFM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Solving+One+Step+Inequalities+by+Adding+and+Subtracting+(Variable+Left+Side)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the inequality and write the answer using interval notation: [latex]-x+4<\\frac{1}{2}x+1[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q703883\">Show Solution<\/button><\/p>\n<div id=\"q703883\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(2,\\infty \\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Solve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q524889\">Show Solution<\/button><\/p>\n<div id=\"q524889\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left[-\\frac{3}{14},\\infty \\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Compound Inequalities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A compound inequality combines two inequalities in one statement<\/li>\n<li class=\"whitespace-normal break-words\">Can be connected by &#8220;and&#8221; (conjunction) or &#8220;or&#8221; (disjunction)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Compound Inequalities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">&#8220;And&#8221; Compound Inequality: Both conditions must be true (e.g., [latex]a < x < b[\/latex])\n\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represent a range of values between two bounds<\/li>\n<li class=\"whitespace-normal break-words\">Solution is the intersection of the two individual inequalities<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">&#8220;Or&#8221; Compound Inequality: At least one condition must be true (e.g., [latex]x < a \\text{ or } x > b[\/latex])\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Represent values satisfying at least one of the conditions<\/li>\n<li class=\"whitespace-normal break-words\">Solution is the union of the two individual inequalities<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Separate into two inequalities and solve individually<\/li>\n<li class=\"whitespace-normal break-words\">Solve the compound inequality intact, operating on all parts simultaneously<\/li>\n<li class=\"whitespace-normal break-words\">Apply the same operation to all parts of the inequality<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to inequality direction when multiplying or dividing by negatives<\/li>\n<li class=\"whitespace-normal break-words\">Combine results for the final solution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the compound inequality [latex]4 < 2x - 8\\le 10[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q265531\">Show Solution<\/button><\/p>\n<div id=\"q265531\" class=\"hidden-answer\" style=\"display: none\">[latex]6 < x\\le 9\\text{ }\\text{ }\\text{or}\\left(6,9\\right][\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the compound inequality: [latex]3y < 4 - 5y < 5+3y[\/latex].\n\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q661493\">Show Solution<\/button><\/p>\n<div id=\"q661493\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(-\\frac{1}{8},\\frac{1}{2}\\right)[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":29,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex: Graph Basic Inequalities and Express Using Interval Notation \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=X0xrHKgbDT0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side) \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=1Z22Xh66VFM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side) 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