{"id":36474,"date":"2017-12-27T18:24:29","date_gmt":"2017-12-27T18:24:29","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=36474"},"modified":"2026-03-28T11:48:05","modified_gmt":"2026-03-28T16:48:05","slug":"compound-inequalities","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/compound-inequalities\/","title":{"rendered":"Solving Compound Inequalities"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_NFVWxQVvej0\" 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_NFVWxQVvej0\" data-source-videoID=\"NFVWxQVvej0\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Solving Compound Inequalities Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Solving Compound Inequalities\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_NFVWxQVvej0:hover {cursor:pointer;} img#videoThumbnailImage_NFVWxQVvej0 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1809-thumb-final-2.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_NFVWxQVvej0\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_NFVWxQVvej0\" 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 Compound Inequalities\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_NFVWxQVvej0\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_NFVWxQVvej0\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_NFVWxQVvej0\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction Nk4_Function() {\n  var x = document.getElementById(\"Nk4\");\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=\"Nk4_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=\"Nk4\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_a_Compound_Inequality\" class=\"smooth-scroll\">What is a Compound Inequality<\/a>\n<ul><\/li>\n<li class=\"toc-h3\"><a href=\"#%E2%80%9COr%E2%80%9D_Inequality\" class=\"smooth-scroll\">\u201cOr\u201d Inequality<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#%E2%80%9CAnd%E2%80%9D_Inequality\" class=\"smooth-scroll\">\u201cAnd\u201d Inequality<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"toc-h2\"><a href=\"#Compound_Inequality_Examples\" class=\"smooth-scroll\">Compound Inequality Examples<\/a>\n<ul><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_2\" class=\"smooth-scroll\">Example #2<\/a><\/li>\n<\/ul>\n<\/li>\n<li class=\"toc-h2\"><a href=\"#Solving_Compound_Inequalities\" class=\"smooth-scroll\">Solving Compound Inequalities<\/a>\n<ul><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_1_1\" class=\"smooth-scroll\">Example #1<\/a><\/li>\n<li class=\"toc-h3\"><a href=\"#Example_2_1\" class=\"smooth-scroll\">Example #2<\/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>Hey guys! Today, we are going to dive into the topic of compound inequalities.<\/p>\n<h2><span id=\"What_is_a_Compound_Inequality\" class=\"m-toc-anchor\"><\/span>What is a Compound Inequality<\/h2>\n<p>\nA compound inequality contains at least two inequalities and is separated either by an \u201cor\u201d or an \u201cand.\u201d Compound inequalities can be used to describe real-world situations, such as the years that a person is not working.<\/p>\n<h3>&#8220;Or&#8221; Inequality<\/h3>\n<p>\nFor example, most people are not working below age 18, or above age 65. This can be represented by the compound inequality \\(x\\lt 18\\text{ or}\\text{ }x\\gt 65\\). This states that people (\\(x\\)) generally will not be working at an age that is less than 18, OR an age that is greater than 65.<\/p>\n<p>This is an example of an \u201cor\u201d inequality. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/01\/image_267.png\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\"  role=\"img\"\/><\/p>\n<h3>&#8220;And&#8221; Inequality<\/h3>\n<p>\nAn example of an \u201cand\u201d compound inequality could be used to describe something like the years a person is in school. This would generally be between the ages of 5 and 25, and we can represent this scenario with the compound inequality \\(5\\leq x\\leq 25\\). This would be read as \u201c<em>x<\/em> is greater than or equal to 5, AND <em>x<\/em> is less than or equal to 25.\u201d In this case, our solution set has to satisfy both constraints. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/01\/image_266.png\" alt=\"\" width=\"\" height=\"\" class=\"aligncenter size-full wp-image-215971\"  role=\"img\" \/><\/p>\n<p>Remember, compound inequalities can be represented as either an \u201cor\u201d statement or an \u201cand\u201d statement, and can incorporate the following symbols: <\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 40%;\">\n<tbody>\n<tr>\n<td>\\(\\gt\\)<\/td>\n<td>Greater than<\/td>\n<\/tr>\n<tr>\n<td>\\(\\lt\\)<\/td>\n<td>Less than<\/td>\n<\/tr>\n<tr>\n<td>\\(\\geq\\)<\/td>\n<td>Greater than or equal to<\/td>\n<\/tr>\n<tr>\n<td>\\(\\leq\\)<\/td>\n<td>Less than or equal to<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Compound_Inequality_Examples\" class=\"m-toc-anchor\"><\/span>Compound Inequality Examples<\/h2>\n<p>\nLet\u2019s look at some examples.<\/p>\n<h3><span id=\"Example_1_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nExample of an \u201cor\u201d compound inequality: \\(6\\lt x\\text{ or}\\text{ }x\\leq -8\\)<\/p>\n<p>This would mean that our solution set is any value greater than 6 or <em>less<\/em> than or equal to -8. Any value that satisfies one of these constraints is considered a solution. <\/p>\n<h3><span id=\"Example_2_1\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nExample of an \u201cand\u201d compound inequality: \\(45\\lt x\\lt 49\\)<\/p>\n<p>This would mean that our solution set is any value greater than 45 and less than 49. All values that fall between these two numbers will satisfy the constraints. <\/p>\n<h2><span id=\"Solving_Compound_Inequalities\" class=\"m-toc-anchor\"><\/span>Solving Compound Inequalities<\/h2>\n<p>\nNow that we have an understanding of what a compound inequality is and what it represents, we can dive into the process for solving compound inequalities. <\/p>\n<h3><span id=\"Example_1_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nLet\u2019s solve for a compound inequality that is separated by an \u201cor\u201d for our first example.<\/p>\n<p>Solve for \\(>z\\): \\(5z+7\\lt 27\\text{ or}\\text{ -}3z\\geq 18\\)<\/p>\n<p>Our first step is to solve each constraint separately. The solving process is the same process for solving multi-step equations. The same rules apply, with one exception.<\/p>\n<p>When you multiply or divide by a negative, the inequality will flip. We will see this process unfold as we finish this example.<\/p>\n<p>So let\u2019s start by solving this inequality.<\/p>\n<div class=\"examplesentence\">\\(5z+7\\gt 27\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe\u2019ll start by subtracting 7 from both sides.<\/p>\n<div class=\"examplesentence\">\\(5z+7-7\\gt 27-7\\)<\/div>\n<p>\n&nbsp;<br \/>\nThat will give us:<\/p>\n<div class=\"examplesentence\">\\(5z\\gt 20\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd then we divide by 5 on both sides.<\/p>\n<div class=\"examplesentence\">\\(\\frac{5z}{5}\\gt \\frac{20}{5}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo \\(z>4\\).<\/p>\n<p>Now, let\u2019s move to our other inequality.<\/p>\n<div class=\"examplesentence\">\\(-3z\\geq 18\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>For this, we just need to divide by -3 on both sides.<\/p>\n<div class=\"examplesentence\">\\(\\frac{-3z}{-3}\\geq \\frac{18}{-3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, when we divide by a negative, our sign flips. So, instead of \\(\\geq\\)  , we&#8217;re going to have . \\(\\leq 18\\text{ }\\div \\left ( -3 \\right )=-6\\).<\/p>\n<div class=\"examplesentence\">\\(z\\leq -6\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo our final answer is:<\/p>\n<div class=\"examplesentence\">\\(z\\lt 4\\text{ or}\\text{ z}\\leq -6\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis is now in a graphable form that can be represented on a line graph.<\/p>\n<p>Now let\u2019s check out a compound inequality that is separated by an \u201cand.\u201d<\/p>\n<p>Solve for <em>x<\/em>: \\(-12\\lt 2-5x\\leq 7\\)<\/p>\n<p>So for an &#8220;and&#8221; inequality like this, we can solve for x by doing inverse operations to all three parts. So, let me show you what I mean.<\/p>\n<div class=\"examplesentence\">\\(-12\\lt 2-5x\\leq 7\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe first thing we want to do is subtract 2 to get <em>x<\/em> by itself. So we&#8217;re going to subtract 2 from this middle part here.<\/p>\n<div class=\"examplesentence\">\\(-12\\lt 2-5x-2\\leq 7\\)<\/div>\n<p>\n&nbsp;<br \/>\nBut if we subtract it from the middle part, we also need to do it to the left and right sides. So we&#8217;ll subtract 2 over here, and we&#8217;ll subtract 2 over here.<\/p>\n<div class=\"examplesentence\">\\(-12-2\\lt 2-5x-2\\leq 7-2\\)<\/div>\n<p>\n&nbsp;<br \/>\n\\(-12-2=-14\\). The twos cancel out and we&#8217;re left with, \\(-5x\\). And \\(7-2=5\\).<\/p>\n<div class=\"examplesentence\">\\(-14\\lt -5x\\leq 5\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, we&#8217;re going to do the same thing but divide by negative 5 on all three parts.<\/p>\n<div class=\"examplesentence\">\\(\\frac{-14}{5}\\lt  \\frac{-5x}{-5}\\leq \\frac{5}{-5}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, when we divide by a negative, we flip our inequality signs. So we\u2019re going to have:<\/p>\n<div class=\"examplesentence\">\\(\\frac{14}{5}\\lt x\\lt -1\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo our answer for this inequality is \\(x\\geq -1\\text{ and}\\text{ x}\\lt \\frac{14}{5}\\).<\/p>\n<h3><span id=\"Example_2_1\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nThere are times when you will be working through the process of solving a compound inequality and your result will be no solution. What does this mean? What would this look like? Let\u2019s take a look.<\/p>\n<p>Solve for \\(x\\): \\(5x-3\\lt 12\\text{ and}\\text{ }4x+1\\gt 25\\)<\/p>\n<p>So we&#8217;re going to start by solving each one of these inequalities separately. <\/p>\n<div class=\"examplesentence\">\\(5x-3\\lt 12\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo let&#8217;s start by adding 3 to both sides of this inequality. <\/p>\n<div class=\"examplesentence\">\\(5x-3+3\\lt 12+3\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis gives us:<\/p>\n<div class=\"examplesentence\">\\(5x\\lt 15\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd divide by 5 on both sides to get:<\/p>\n<div class=\"examplesentence\">\\(x\\lt 3\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow let\u2019s move to this inequality over here:<\/p>\n<div class=\"examplesentence\">\\(4x+1\\gt 25\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe\u2019re going to subtract 1 from both sides.<\/p>\n<div class=\"examplesentence\">\\(4x+1-1\\gt 25-1\\)<\/div>\n<p>\n&nbsp;<br \/>\nThat gives us:<\/p>\n<div class=\"examplesentence\">\\(4x\\gt 24\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd then divide by 4 on both sides.<\/p>\n<div class=\"examplesentence\">\\(\\frac{4x}{4}\\gt  \\frac{24}{4}\\)<br \/>\n\\(x\\gt 6\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo our solution is \\(x\\lt 3\\) and \\(x\\gt 6\\).<\/p>\n<p>But wait, this is impossible. We are unable to find a value for \\(x\\) that is both less than 3, and greater than 6. If this occurs, the compound inequality is said to have \u201cno solution.\u201d<\/p>\n<p>And that\u2019s all there is to it. I hope that this video was helpful for you! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":111003,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-36474","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-inequalities-videos","7":"page_type-video","8":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/36474","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=36474"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/36474\/revisions"}],"predecessor-version":[{"id":281099,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/36474\/revisions\/281099"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/111003"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=36474"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}