{"id":1317,"date":"2025-07-24T04:18:02","date_gmt":"2025-07-24T04:18:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1317"},"modified":"2026-03-09T07:24:20","modified_gmt":"2026-03-09T07:24:20","slug":"absolute-value-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/absolute-value-functions-fresh-take\/","title":{"raw":"Absolute Value Functions: Fresh Take","rendered":"Absolute Value Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph an absolute value function.<\/li>\r\n \t<li>Solve an absolute value equation.<\/li>\r\n \t<li>Solve an absolute value inequality.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Understanding Absolute Value<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Absolute value measures distance from zero on the number line. Think of it like this: if you're standing at zero and you walk 5 steps in either direction, you've traveled 5 steps. That's why both [latex]|5| = 5[\/latex] and [latex]|-5| = 5[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since distance is never negative, absolute value is never negative. This one idea unlocks everything about absolute value functions, equations, and inequalities.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Mathematically, we can define absolute value as:<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]|x| = \\begin{cases} x &amp; \\text{if } x \\geq 0 \\ -x &amp; \\text{if } x &lt; 0 \\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-egefebdf-LnfhdtjcpVY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LnfhdtjcpVY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-egefebdf-LnfhdtjcpVY\" 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=14660068&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-egefebdf-LnfhdtjcpVY&vembed=0&video_id=LnfhdtjcpVY&video_target=tpm-plugin-egefebdf-LnfhdtjcpVY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/What+is+Absolute+Value%3F+%7C+Absolute+Value+Examples+%7C+Math+with+Mr.+J_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is Absolute Value? | Absolute Value Examples | Math with Mr. J\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Graphing Absolute Value Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The parent absolute value function [latex]f(x) = |x|[\/latex] creates a distinctive V-shape with its vertex (the pointy corner) at the origin [latex](0, 0)[\/latex]. All absolute value functions have this V-shape, though the vertex can move and the V can flip, stretch, or compress.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The basic graph opens upward with the vertex at [latex](0, 0)[\/latex]. Each side of the V has a slope of 1 or -1, creating a perfect symmetric V.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Recall:<\/strong> Transformations work the same way for absolute value functions as they do for other functions. The general form [latex]f(x) = a|x - h| + k[\/latex] tells you:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The vertex is at [latex](h, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]a &gt; 0[\/latex], the V opens upward<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]a &lt; 0[\/latex], the V opens downward (it's flipped)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The value of [latex]|a|[\/latex] tells you how steep the sides are<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gddbfdeb-weGn1iTnm-I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/weGn1iTnm-I?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gddbfdeb-weGn1iTnm-I\" 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=14660069&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gddbfdeb-weGn1iTnm-I&vembed=0&video_id=weGn1iTnm-I&video_target=tpm-plugin-gddbfdeb-weGn1iTnm-I'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Graphing+Absolute+Value+Functions+(y%3Da%7Cx-h%7C%2Bk)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Absolute Value Functions (y=a|x-h|+k)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Graph [latex]g(x) = 2|x - 3| + 1[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"abs1\"]Show Solution[\/reveal-answer] [hidden-answer a=\"abs1\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Start by identifying the transformations:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]h = 3[\/latex], so shift right 3 units<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]k = 1[\/latex], so shift up 1 unit<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]a = 2[\/latex], so vertical stretch by a factor of 2<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The vertex is at [latex](3, 1)[\/latex]. Since [latex]a = 2 &gt; 0[\/latex], the graph opens upward and is steeper than the parent function.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">The vertex is your starting point. Find [latex](h, k)[\/latex] first, plot that point, then determine which direction the V opens and how steep it is.<\/section>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Solving Absolute Value Equations<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">When you see [latex]|A| = B[\/latex], you're asking: \"What has a distance of B from zero?\" There are usually two answers: the positive and negative versions. That's why [latex]|x| = 5[\/latex] has two solutions: [latex]x = 5[\/latex] and [latex]x = -5[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">However, if [latex]B &lt; 0[\/latex] (if B is negative), there's no solution because distance can't be negative!<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving [latex]|A| = B[\/latex]<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Check if [latex]B &lt; 0[\/latex]. If yes, there's no solution.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Isolate the absolute value expression on one side of the equation.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Write two separate equations: [latex]A = B[\/latex] and [latex]A = -B[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solve each equation.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Check both solutions in the original equation.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbfbdgcg-_cHbhzQVd7Y\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_cHbhzQVd7Y?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fbfbdgcg-_cHbhzQVd7Y\" 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=14660070&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fbfbdgcg-_cHbhzQVd7Y&vembed=0&video_id=_cHbhzQVd7Y&video_target=tpm-plugin-fbfbdgcg-_cHbhzQVd7Y'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+To+Solve+Absolute+Value+Equations%2C+Basic+Introduction%2C+Algebra_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Solve Absolute Value Equations, Basic Introduction, Algebra\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|4x - 1| + 3 = 10[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"abs2\"]Show Solution[\/reveal-answer] [hidden-answer a=\"abs2\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex]\\begin{align} \\text{Original equation} &amp;: |4x - 1| + 3 = 10 \\ \\text{Isolate absolute value} &amp;: |4x - 1| = 7 \\ \\text{Case 1: } 4x - 1 &amp;= 7 &amp; \\text{Case 2: } 4x - 1 &amp;= -7 \\ 4x &amp;= 8 &amp; 4x &amp;= -6 \\ x &amp;= 2 &amp; x &amp;= -\\frac{3}{2} \\end{align}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The solutions are [latex]x = 2[\/latex] and [latex]x = -\\frac{3}{2}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|2x + 5| = -3[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"abs3\"]Show Solution[\/reveal-answer] [hidden-answer a=\"abs3\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">No solution.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since absolute value represents distance, it cannot equal a negative number. The equation [latex]|2x + 5| = -3[\/latex] has no solution.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">If the equation simplifies to [latex]|A| = 0[\/latex], there's only ONE solution. Set [latex]A = 0[\/latex] and solve. Zero is the only number that's exactly zero units from zero!<\/section>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Solving Absolute Value Inequalities<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Absolute value inequalities ask about ranges of distance rather than exact distances. The key is understanding what \"less than\" versus \"greater than\" means for distance:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]|x| &lt; 5[\/latex] means \"distance from zero is less than 5\" \u2192 x is between -5 and 5<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]|x| &gt; 5[\/latex] means \"distance from zero is greater than 5\" \u2192 x is less than -5 OR greater than 5<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Think of it this way: \"less than\" keeps you close (between two values), while \"greater than\" pushes you far away (outside the range).<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gbfcbaad-30PrH3OIiqk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/30PrH3OIiqk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gbfcbaad-30PrH3OIiqk\" 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=14660071&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gbfcbaad-30PrH3OIiqk&vembed=0&video_id=30PrH3OIiqk&video_target=tpm-plugin-gbfcbaad-30PrH3OIiqk'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+To+Solve+Absolute+Value+Inequalities%2C+Basic+Introduction%2C+Algebra_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Solve Absolute Value Inequalities, Basic Introduction, Algebra\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving [latex]|A| &lt; B[\/latex] or [latex]|A| \\leq B[\/latex]<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Isolate the absolute value expression.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Rewrite as a compound inequality: [latex]-B &lt; A &lt; B[\/latex] (or [latex]-B \\leq A \\leq B[\/latex]).<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solve the compound inequality for the variable.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Write your answer in interval notation.<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving [latex]|A| &gt; B[\/latex] or [latex]|A| \\geq B[\/latex]<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Isolate the absolute value expression.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Write two separate inequalities: [latex]A &lt; -B[\/latex] OR [latex]A &gt; B[\/latex] (adjust inequality symbols if needed).<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solve each inequality separately.<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Write your answer as the union of two intervals using [latex]\\cup[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|x + 2| \\leq 6[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"abs4\"]Show Solution[\/reveal-answer] [hidden-answer a=\"abs4\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex]\\begin{align} \\text{Original inequality} &amp;: |x + 2| \\leq 6 \\ \\text{Compound inequality} &amp;: -6 \\leq x + 2 \\leq 6 \\ \\text{Subtract 2 from all parts} &amp;: -6 - 2 \\leq x \\leq 6 - 2 \\ \\text{Simplify} &amp;: -8 \\leq x \\leq 4 \\ \\text{Interval notation} &amp;: [-8, 4] \\end{align}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The solution is all x-values between -8 and 4, including the endpoints.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|3x - 1| &gt; 8[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[reveal-answer q=\"abs5\"]Show Solution[\/reveal-answer] [hidden-answer a=\"abs5\"]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">[latex]\\begin{align} \\text{Original inequality} &amp;: |3x - 1| &gt; 8 \\ \\text{Case 1: } 3x - 1 &amp;&gt; 8 &amp; \\text{Case 2: } 3x - 1 &amp;&lt; -8 \\ 3x &amp;&gt; 9 &amp; 3x &amp;&lt; -7 \\ x &amp;&gt; 3 &amp; x &amp;&lt; -\\frac{7}{3} \\end{align}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The solution is [latex]x &lt; -\\frac{7}{3}[\/latex] or [latex]x &gt; 3[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In interval notation: [latex]\\left(-\\infty, -\\frac{7}{3}\\right) \\cup (3, \\infty)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[\/hidden-answer]<\/p>\r\n\r\n<\/section>&nbsp;\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The union symbol [latex]\\cup[\/latex] combines two separate solution sets. We use it when solutions come in two pieces that don't connect.<\/section>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Visualizing with Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">You can verify solutions to absolute value inequalities by graphing. Convert your inequality to a two-variable form and graph it\u2014the x-intercepts show the boundary points of your solution, and the shaded region shows which x-values work.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For example, to graph [latex]|x - 2| &lt; 5[\/latex]:<\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Rewrite as [latex]y &gt; |x - 2| - 5[\/latex] (move everything to one side)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Graph the inequality using a graphing calculator<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The x-intercepts are at [latex]x = -3[\/latex] and [latex]x = 7[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The shaded region confirms your solution [latex](-3, 7)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-achffcae-KRmRSaHP2-k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KRmRSaHP2-k?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-achffcae-KRmRSaHP2-k\" 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=14660072&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-achffcae-KRmRSaHP2-k&vembed=0&video_id=KRmRSaHP2-k&video_target=tpm-plugin-achffcae-KRmRSaHP2-k'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Desmos+-+Graphing+Absolute+Value+Inequalities+and+their+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDesmos: Graphing Absolute Value Inequalities and their Solutions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph an absolute value function.<\/li>\n<li>Solve an absolute value equation.<\/li>\n<li>Solve an absolute value inequality.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Understanding Absolute Value<\/h2>\n<div class=\"textbox shaded\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Absolute value measures distance from zero on the number line. Think of it like this: if you&#8217;re standing at zero and you walk 5 steps in either direction, you&#8217;ve traveled 5 steps. That&#8217;s why both [latex]|5| = 5[\/latex] and [latex]|-5| = 5[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Since distance is never negative, absolute value is never negative. This one idea unlocks everything about absolute value functions, equations, and inequalities.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Mathematically, we can define absolute value as:<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]|x| = \\begin{cases} x & \\text{if } x \\geq 0 \\ -x & \\text{if } x < 0 \\end{cases}[\/latex]<\/p>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-egefebdf-LnfhdtjcpVY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/LnfhdtjcpVY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-egefebdf-LnfhdtjcpVY\" 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=14660068&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-egefebdf-LnfhdtjcpVY&#38;vembed=0&#38;video_id=LnfhdtjcpVY&#38;video_target=tpm-plugin-egefebdf-LnfhdtjcpVY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/What+is+Absolute+Value%3F+%7C+Absolute+Value+Examples+%7C+Math+with+Mr.+J_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is Absolute Value? | Absolute Value Examples | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Graphing Absolute Value Functions<\/h2>\n<div class=\"textbox shaded\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The parent absolute value function [latex]f(x) = |x|[\/latex] creates a distinctive V-shape with its vertex (the pointy corner) at the origin [latex](0, 0)[\/latex]. All absolute value functions have this V-shape, though the vertex can move and the V can flip, stretch, or compress.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The basic graph opens upward with the vertex at [latex](0, 0)[\/latex]. Each side of the V has a slope of 1 or -1, creating a perfect symmetric V.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Recall:<\/strong> Transformations work the same way for absolute value functions as they do for other functions. The general form [latex]f(x) = a|x - h| + k[\/latex] tells you:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The vertex is at [latex](h, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]a > 0[\/latex], the V opens upward<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]a < 0[\/latex], the V opens downward (it&#8217;s flipped)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The value of [latex]|a|[\/latex] tells you how steep the sides are<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gddbfdeb-weGn1iTnm-I\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/weGn1iTnm-I?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gddbfdeb-weGn1iTnm-I\" 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=14660069&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gddbfdeb-weGn1iTnm-I&#38;vembed=0&#38;video_id=weGn1iTnm-I&#38;video_target=tpm-plugin-gddbfdeb-weGn1iTnm-I\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Graphing+Absolute+Value+Functions+(y%3Da%7Cx-h%7C%2Bk)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraphing Absolute Value Functions (y=a|x-h|+k)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Graph [latex]g(x) = 2|x - 3| + 1[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qabs1\">Show Solution<\/button> <\/p>\n<div id=\"qabs1\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Start by identifying the transformations:<\/p>\n<ul class=\"&#091;li_&amp;&#093;:mb-0 &#091;li_&amp;&#093;:mt-1.5 &#091;li_&amp;&#093;:gap-1.5 &#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">[latex]h = 3[\/latex], so shift right 3 units<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]k = 1[\/latex], so shift up 1 unit<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]a = 2[\/latex], so vertical stretch by a factor of 2<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The vertex is at [latex](3, 1)[\/latex]. Since [latex]a = 2 > 0[\/latex], the graph opens upward and is steeper than the parent function.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The vertex is your starting point. Find [latex](h, k)[\/latex] first, plot that point, then determine which direction the V opens and how steep it is.<\/section>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Solving Absolute Value Equations<\/h2>\n<div class=\"textbox shaded\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">When you see [latex]|A| = B[\/latex], you&#8217;re asking: &#8220;What has a distance of B from zero?&#8221; There are usually two answers: the positive and negative versions. That&#8217;s why [latex]|x| = 5[\/latex] has two solutions: [latex]x = 5[\/latex] and [latex]x = -5[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">However, if [latex]B < 0[\/latex] (if B is negative), there&#8217;s no solution because distance can&#8217;t be negative!<\/p>\n<\/div>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving [latex]|A| = B[\/latex]<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Check if [latex]B < 0[\/latex]. If yes, there&#8217;s no solution.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Isolate the absolute value expression on one side of the equation.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Write two separate equations: [latex]A = B[\/latex] and [latex]A = -B[\/latex].<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Solve each equation.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Check both solutions in the original equation.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fbfbdgcg-_cHbhzQVd7Y\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/_cHbhzQVd7Y?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fbfbdgcg-_cHbhzQVd7Y\" 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=14660070&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fbfbdgcg-_cHbhzQVd7Y&#38;vembed=0&#38;video_id=_cHbhzQVd7Y&#38;video_target=tpm-plugin-fbfbdgcg-_cHbhzQVd7Y\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+To+Solve+Absolute+Value+Equations%2C+Basic+Introduction%2C+Algebra_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Solve Absolute Value Equations, Basic Introduction, Algebra\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|4x - 1| + 3 = 10[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qabs2\">Show Solution<\/button> <\/p>\n<div id=\"qabs2\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{align} \\text{Original equation} &: |4x - 1| + 3 = 10 \\ \\text{Isolate absolute value} &: |4x - 1| = 7 \\ \\text{Case 1: } 4x - 1 &= 7 & \\text{Case 2: } 4x - 1 &= -7 \\ 4x &= 8 & 4x &= -6 \\ x &= 2 & x &= -\\frac{3}{2} \\end{align}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The solutions are [latex]x = 2[\/latex] and [latex]x = -\\frac{3}{2}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|2x + 5| = -3[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qabs3\">Show Solution<\/button> <\/p>\n<div id=\"qabs3\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">No solution.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">Since absolute value represents distance, it cannot equal a negative number. The equation [latex]|2x + 5| = -3[\/latex] has no solution.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">If the equation simplifies to [latex]|A| = 0[\/latex], there&#8217;s only ONE solution. Set [latex]A = 0[\/latex] and solve. Zero is the only number that&#8217;s exactly zero units from zero!<\/section>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Solving Absolute Value Inequalities<\/h2>\n<div class=\"textbox shaded\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Absolute value inequalities ask about ranges of distance rather than exact distances. The key is understanding what &#8220;less than&#8221; versus &#8220;greater than&#8221; means for distance:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">[latex]|x| < 5[\/latex] means &#8220;distance from zero is less than 5&#8221; \u2192 x is between -5 and 5<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]|x| > 5[\/latex] means &#8220;distance from zero is greater than 5&#8221; \u2192 x is less than -5 OR greater than 5<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Think of it this way: &#8220;less than&#8221; keeps you close (between two values), while &#8220;greater than&#8221; pushes you far away (outside the range).<\/p>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gbfcbaad-30PrH3OIiqk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/30PrH3OIiqk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gbfcbaad-30PrH3OIiqk\" 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=14660071&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gbfcbaad-30PrH3OIiqk&#38;vembed=0&#38;video_id=30PrH3OIiqk&#38;video_target=tpm-plugin-gbfcbaad-30PrH3OIiqk\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+To+Solve+Absolute+Value+Inequalities%2C+Basic+Introduction%2C+Algebra_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow To Solve Absolute Value Inequalities, Basic Introduction, Algebra\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving [latex]|A| < B[\/latex] or [latex]|A| \\leq B[\/latex]<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Isolate the absolute value expression.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Rewrite as a compound inequality: [latex]-B < A < B[\/latex] (or [latex]-B \\leq A \\leq B[\/latex]).<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Solve the compound inequality for the variable.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Write your answer in interval notation.<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving [latex]|A| > B[\/latex] or [latex]|A| \\geq B[\/latex]<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Isolate the absolute value expression.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Write two separate inequalities: [latex]A < -B[\/latex] OR [latex]A > B[\/latex] (adjust inequality symbols if needed).<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Solve each inequality separately.<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Write your answer as the union of two intervals using [latex]\\cup[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|x + 2| \\leq 6[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qabs4\">Show Solution<\/button> <\/p>\n<div id=\"qabs4\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{align} \\text{Original inequality} &: |x + 2| \\leq 6 \\ \\text{Compound inequality} &: -6 \\leq x + 2 \\leq 6 \\ \\text{Subtract 2 from all parts} &: -6 - 2 \\leq x \\leq 6 - 2 \\ \\text{Simplify} &: -8 \\leq x \\leq 4 \\ \\text{Interval notation} &: [-8, 4] \\end{align}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The solution is all x-values between -8 and 4, including the endpoints.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Solve [latex]|3x - 1| > 8[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qabs5\">Show Solution<\/button> <\/p>\n<div id=\"qabs5\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-&#091;1.7&#093;\">[latex]\\begin{align} \\text{Original inequality} &: |3x - 1| > 8 \\ \\text{Case 1: } 3x - 1 &> 8 & \\text{Case 2: } 3x - 1 &< -8 \\ 3x &> 9 & 3x &< -7 \\ x &> 3 & x &< -\\frac{7}{3} \\end{align}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">The solution is [latex]x < -\\frac{7}{3}[\/latex] or [latex]x > 3[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\">In interval notation: [latex]\\left(-\\infty, -\\frac{7}{3}\\right) \\cup (3, \\infty)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-&#091;1.7&#093;\"><\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">The union symbol [latex]\\cup[\/latex] combines two separate solution sets. We use it when solutions come in two pieces that don&#8217;t connect.<\/section>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Visualizing with Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Main Idea<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">You can verify solutions to absolute value inequalities by graphing. Convert your inequality to a two-variable form and graph it\u2014the x-intercepts show the boundary points of your solution, and the shaded region shows which x-values work.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For example, to graph [latex]|x - 2| < 5[\/latex]:<\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1.5 [li_&amp;]:gap-1.5 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-2 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Rewrite as [latex]y > |x - 2| - 5[\/latex] (move everything to one side)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Graph the inequality using a graphing calculator<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The x-intercepts are at [latex]x = -3[\/latex] and [latex]x = 7[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The shaded region confirms your solution [latex](-3, 7)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-achffcae-KRmRSaHP2-k\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KRmRSaHP2-k?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-achffcae-KRmRSaHP2-k\" 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=14660072&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-achffcae-KRmRSaHP2-k&#38;vembed=0&#38;video_id=KRmRSaHP2-k&#38;video_target=tpm-plugin-achffcae-KRmRSaHP2-k\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Desmos+-+Graphing+Absolute+Value+Inequalities+and+their+Solutions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDesmos: Graphing Absolute Value Inequalities and their Solutions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":35,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"What is Absolute Value? | Absolute Value Examples | Math 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