{"id":660,"date":"2013-05-28T15:02:08","date_gmt":"2013-05-28T15:02:08","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=660"},"modified":"2026-03-26T10:10:17","modified_gmt":"2026-03-26T15:10:17","slug":"solving-absolute-value-equations","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/solving-absolute-value-equations\/","title":{"rendered":"Solving Absolute Value Equations"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_7XdqDdXlRbo\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_7XdqDdXlRbo\" data-source-videoID=\"7XdqDdXlRbo\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Solving Absolute Value Equations Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Solving Absolute Value Equations\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_7XdqDdXlRbo:hover {cursor:pointer;} img#videoThumbnailImage_7XdqDdXlRbo {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1309-thumbnail-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_7XdqDdXlRbo\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_7XdqDdXlRbo\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Solving Absolute Value Equations\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_7XdqDdXlRbo\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_7XdqDdXlRbo\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_7XdqDdXlRbo\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction n1I_Function() {\n  var x = document.getElementById(\"n1I\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"n1I_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"n1I\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_Absolute_Value\" class=\"smooth-scroll\">What is Absolute Value?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_2\" class=\"smooth-scroll\">Example #2<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_3\" class=\"smooth-scroll\">Example #3<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_4\" class=\"smooth-scroll\">Example #4<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Example_5\" class=\"smooth-scroll\">Example #5<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video on solving absolute value equations.<\/p>\n<h2><span id=\"What_is_Absolute_Value\" class=\"m-toc-anchor\"><\/span>What is Absolute Value?<\/h2>\n<p>\nBefore we get started, let\u2019s review what absolute value is. The <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/absolute-value\/\">absolute value<\/a> of a number is the distance that number is away from zero. In other words, you make any number positive. \\(|27|=27\\), but \\(|-27|=27\\) also. You make any number inside the absolute value signs positive.<\/p>\n<p>When we solve absolute value equations, our process will look very similar to solving <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/solving-multi-step-equations\/\">regular equations<\/a>. There is however, one major difference. When solving absolute value equations, we need to consider when the expression inside the absolute value bars is positive and when it is negative.<\/p>\n<h2><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h2>\n<p>\nLet\u2019s look at a few examples.<\/p>\n<div class=\"examplesentence\">\\(4|x|=24\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo we want to solve this for \\(x\\).<\/p>\n<p>We&#8217;re going to start by dividing both sides of the equation by 4. The reason we do this is so that we can isolate our absolute value expression. You always want to isolate your absolute value part first.<\/p>\n<div class=\"examplesentence\" style=\"font-size: 120%;\">\\(\\frac{4|x|}{4}=\\frac{24}{4}\\)<br \/>\n<span style=\"font-size: 90%;\">\\(|x|=6\\)<\/span><\/div>\n<p>\n&nbsp;<br \/>\nThen, remember the definition of absolute value. The absolute value of a number is always positive, so \\(x=\\pm 6\\) because if we take |6|, we\u2019ll get 6, and if we take \\(|-6|\\), we&#8217;ll also get positive 6.<\/p>\n<div class=\"examplesentence\">\\(x=\\pm 6\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo that&#8217;s how we solve a simple absolute value equation.<\/p>\n<h2><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h2>\n<p>\nLet&#8217;s try another one.<\/p>\n<div class=\"examplesentence\">\\(|2x+3|=21\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis time we have an expression inside our absolute value bars, so we&#8217;re going to solve for \\(x\\) in a very similar way. However, we don&#8217;t have to get the absolute value expression by itself this time because it&#8217;s already isolated. So, we&#8217;re going to start by getting rid of the absolute value bars. Now, the way that we do this is we split our expression inside into two different equations.<\/p>\n<p>We want to consider this to be a positive answer and consider this to be a negative answer. Remember, last time we had \\(|x|=6\\), and we knew that \\(x=6\\pm\\). \\(x\\) was our expression inside the absolute value bars, and we set that equal to positive and negative of the answer. So we&#8217;re going to do the same thing.<\/p>\n<div class=\"examplesentence\">\\(2x+3=21~~~~~~~~2x+3=-21\\)<\/div>\n<\/blockquote>\n<p>\n&nbsp;<br \/>\nWe do this because the absolute value symbols make any answer positive, so the inside expression could actually be either positive or negative.<\/p>\n<p>Now, all we do is solve both equations for \\(x\\). And since we have the same expressions on both sides, we&#8217;ll follow the same steps for both equations.<\/p>\n<div class=\"equations-container\">\n<div class=\"equation-column\">\n<div class=\"equation\">\n<div class=\"lhs\">\\(2x + 3\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(21\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(2x + 3 &#8211; 3\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(21 &#8211; 3\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(2x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(18\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(\\frac{2x}{2}\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{18}{2}\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(9\\)<\/div>\n<\/div>\n<\/div>\n<div class=\"equation-column\">\n<div class=\"equation\">\n<div class=\"lhs\">\\(2x + 3\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-21\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(2x + 3 &#8211; 3\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-21 &#8211; 3\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(2x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-24\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(\\frac{2x}{2}\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{-24}{2}\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-12\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>\n&nbsp;<br \/>\nSince we are considering both a positive and a negative answer, we will get two different values of \\(x\\). Our solutions for this equation are \\(x=9\\) and \\(x=-12\\).<\/p>\n<p>We can check them by plugging them into our original equation.<\/p>\n<div class=\"examplesentence\">\\(|2(9)+3|=|18+3|\\)\\(=|21|=21\\)<br \/>\n\\(|2(-12)+3|=|-24+3|\\)\\(=|-21|=21\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example #3<\/h2>\n<p>\nLet\u2019s try another one.<\/p>\n<div class=\"examplesentence\">\\(|7x-14|-3=25\\)<\/div>\n<p>\n&nbsp;<br \/>\nFor this one, our first step needs to be to get the absolute value expression by itself. So we need to get rid of this minus 3. So to do that, we&#8217;ll add 3 to both sides of the equation.<\/p>\n<div class=\"examplesentence\">\\(|7x-14|-3+3=25+3\\)<br \/>\n\\(|7x-14|=28\\)<\/div>\n<p>\n&nbsp;<br \/>\nFrom here we split our equation into two different ones. Remember, we want to set our expression on the left equal to the positive and negative value of the right side. Then, we&#8217;re just going to solve both of our equations like we did last time.<\/p>\n<div class=\"equations-container\">\n<div class=\"equation-column\">\n<div class=\"equation\">\n<div class=\"lhs\">\\(7x &#8211; 14\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(28\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(7x &#8211; 14 + 14\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(28 + 14\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(7x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(42\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(\\frac{7x}{7}\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{42}{7}\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(6\\)<\/div>\n<\/div>\n<\/div>\n<div class=\"equation-column\">\n<div class=\"equation\">\n<div class=\"lhs\">\\(7x &#8211; 14\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-28\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(7x &#8211; 14 + 14\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-28 + 14\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(7x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-14\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(\\frac{7x}{7}\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{-14}{7}\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-2\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>\n&nbsp;<br \/>\nSo our answers for this equation are \\(x=6\\) and \\(x=-2\\).<\/p>\n<h2><span id=\"Example_4\" class=\"m-toc-anchor\"><\/span>Example #4<\/h2>\n<p>\nLet\u2019s for a second consider one of the problems we might encounter when working with these equations. Take a look at this example:<\/p>\n<div class=\"examplesentence\">\\(|2x-15|+3=-25\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe first thing we would do in this example is subtract 3 from both sides, but when we do that, we get this:<\/p>\n<div class=\"examplesentence\">\\(|2x-15|+3-3=-25-3\\)<br \/>\n\\(|2x-15|=-28\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhat\u2019s wrong with this equation? Well, we know that absolute value signs always make a number positive, so it is not <em>possible<\/em> to have an absolute value expression equal to a negative number. If you ever encounter this, just know that it means there are <em>no solutions<\/em> to the equation. So if you get a negative number on the right side, when there&#8217;s an absolute value expression on the left, this means there are no solutions.<\/p>\n<h2><span id=\"Example_5\" class=\"m-toc-anchor\"><\/span>Example #5<\/h2>\n<p>\nBefore we go, I want to work through one more example.<\/p>\n<div class=\"examplesentence\">\\(|3x-2|-24=14\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo we&#8217;re going to solve this just like we have been. Start by adding 24 to both sides.<\/p>\n<div class=\"examplesentence\">\\(|3x-2|-24+24=14+24\\)<br \/>\n\\(|3x-2|=38\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, we get rid of the absolute value symbols and split our equation into two different ones.<\/p>\n<div class=\"examplesentence\">\\(3x-2=38~~~~~~~~3x-2=-38\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, we solve both equations for \\(x\\).<\/p>\n<div class=\"equations-container\">\n<div class=\"equation-column\">\n<div class=\"equation\">\n<div class=\"lhs\">\\(3x &#8211; 2\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(38\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(3x &#8211; 2 + 2\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(38 + 2\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(3x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(40\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(\\frac{3x}{3}\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{40}{3}\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{40}{3}\\)<\/div>\n<\/div>\n<\/div>\n<div class=\"equation-column\">\n<div class=\"equation\">\n<div class=\"lhs\">\\(3x &#8211; 2\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-38\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(3x &#8211; 2 + 2\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-38 + 2\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(3x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-36\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(\\frac{3x}{3}\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(\\frac{-36}{3}\\)<\/div>\n<\/div>\n<div class=\"equation\">\n<div class=\"lhs\">\\(x\\)<\/div>\n<div class=\"equal\">\\(=\\)<\/div>\n<div class=\"rhs\">\\(-12\\)<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>\n&nbsp;<br \/>\nSo, our answers are \\(x=\\frac{40}{3}\\) and \\(x=-12\\). Don&#8217;t be afraid by this fraction right here, you can just leave it just like this\u2014so your answer is, \\(x=\\frac{40}{3}\\), and that is correct. So the solutions for this equation again are \\(x=\\frac{40}{3}\\) and \\(x=-12\\).<\/p>\n<p>I hope this review on solving absolute value equations was helpful. 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