{"id":2857,"date":"2024-02-09T19:39:18","date_gmt":"2024-02-09T19:39:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2857"},"modified":"2026-03-09T17:48:49","modified_gmt":"2026-03-09T17:48:49","slug":"6-2-3-solving-compound-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-2-3-solving-compound-inequalities\/","title":{"raw":"6.2.3 Solving Compound Inequalities","rendered":"6.2.3 Solving Compound Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Solve compound inequalities with OR (Unions) - express solutions graphically, in set-builder notation and with interval notation<\/li>\r\n \t<li>Solve compound inequalities with AND (Intersections) - express solutions graphically, in set-builder notation and with interval notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>Key words<\/h1>\r\n<ul>\r\n \t<li><strong>Compound Inequality<\/strong>: two inequalities joined as a union or an intersection<\/li>\r\n \t<li><strong>Union<\/strong>: values exist in either set. Key word is \"or\".<\/li>\r\n \t<li><strong>Intersection<\/strong>: values exist in both sets. Key word is \"and\".<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Compound Inequalities<\/h2>\r\nA <em><strong>compound inequality<\/strong><\/em> is two inequalities joined together with the word \"or\" or the word \"and\". When the two inequalities are joined with \"or\", the solution set must satisfy <em>either<\/em> equation. For example, [latex]x\\lt 2 \\text{ or } x\\ge 5[\/latex] tells us that the [latex]x[\/latex]-values must be either less than [latex]2[\/latex] or greater than or equal to [latex]5[\/latex]. On the other hand, [latex]x\\ge 2\\text{ and }x\\lt 5[\/latex] tells us that\u00a0the [latex]x[\/latex]-values must be greater than or equal to [latex]2[\/latex] and also less than [latex]5[\/latex]. When \"or\" is used, the solution set is a <em><strong>union<\/strong><\/em> of the solution sets of both inequalities. When \"and\" is used, the solution set is an <em><strong>intersection<\/strong><\/em>\u00a0of the solution sets of both inequalities.\r\n<h2>Unions<\/h2>\r\nThe solution of a compound inequality that consists of two inequalities joined with the word <em>or<\/em> is the <strong>union<\/strong> of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.\r\n\r\nThe following example shows how to solve a one-step inequality in the <em>or<\/em> form. Note how each inequality is treated independently until the end, where the solution is described in terms of both inequalities. We will use the same properties to solve compound inequalities that we used to solve single inequalities.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]x[\/latex]. \u00a0[latex]3x\u20131&lt;8[\/latex] <em>or<\/em> [latex]x\u20135&gt;0[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nSolve each inequality by isolating the variable.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x-5&gt;0\\,\\,\\,\\,\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3x-1&lt;8\\,\\,\\\\\\underline{\\,\\,\\,+5\\,\\,+5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,\\,+1}\\\\x\\,\\,&gt;5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{3x}\\,\\,\\,&lt;\\underline{9}\\\\{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{3}\\\\x&lt;3\\,\\,\\,\\\\x&gt;5\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,x&lt;3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nSet-builder notation: [latex]\\{\\;x\\large\\;|\\;\\normalsize x\\lt 3\\text{ or }x\\gt 5,\\;y\\in\\mathbb{R}\\}[\/latex]\r\n\r\nInterval notation: [latex]\\left(-\\infty, 3\\right)\\cup\\left(5,\\infty\\right)[\/latex]\r\n\r\nThe solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first, before writing the solution using interval notation.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182844\/image078.jpg\" alt=\"Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.\" width=\"575\" height=\"53\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]y[\/latex].\u00a0\u00a0[latex]2y+7\\lt13[\/latex] <em>or<\/em> [latex]\u22123y\u20132\\lt10[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nSolve each inequality separately.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}2y+7&lt;13\\,\\,\\,\\,\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-3y-2\\lt 10\\\\\\underline{\\,\\,\\,-7\\,\\,-7}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+2\\,\\,\\,+2}\\\\\\underline{2y}&lt;\\underline{6}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-3y}&lt;\\underline{12}\\\\{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-3}\\\\y&lt;3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\gt -4\\\\y&lt;3\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,y\\gt -4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThe inequality sign is reversed with division by a negative number.\r\n\r\nSince [latex]y[\/latex] could be less than\u00a0[latex]3[\/latex] or greater than [latex]\u22124,\\;y[\/latex]\u00a0could be any real number. Graphing the inequality helps with this interpretation.\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter wp-image-1888\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/09\/28004312\/Union.png\" alt=\"Line graph\" width=\"524\" height=\"117\" \/>\r\n\r\nThe individual graphs show empty dots at [latex]y=3[\/latex] and [latex]y=-4[\/latex], but these graphs combine to all real numbers.\r\n\r\nSet-builder notation: [latex]\\{\\;y\\large\\;|\\;\\normalsize \\;y\\in\\mathbb{R}\\}[\/latex]\r\n\r\nInterval notation: [latex]x\\in\\left(-\\infty,\\infty\\right)[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the last example, the final answer included solutions whose intervals overlapped. This caused the answer to include all numbers on the number line. In words, we call this solution \"all real numbers\".\u00a0Any real number will produce a true statement for either\u00a0[latex]y&lt;3\\text{ or }y\\gt -4[\/latex] when it is substituted for <em>y<\/em>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]z[\/latex].\r\n\r\n[latex]5z\u20133\\gt\u221218[\/latex] <em>or<\/em> [latex]\u22122z\u20131\\gt15[\/latex]\r\n\r\n&nbsp;\r\n<h4><strong>Solution<\/strong><\/h4>\r\nSolve each inequality separately.\u00a0Combine the solutions.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}5z-3&gt;18\\,\\,\\,\\,\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2z-1&gt;15\\\\\\underline{\\,\\,\\,+3\\,\\,\\,+3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,\\,\\,+1}\\\\\\underline{5z}&gt;\\underline{-15}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2z}&gt;\\underline{16}\\\\{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-2}\\\\z&gt;-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z&lt;-8\\\\z&gt;-3\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,z&lt;-8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nSet-builder notation:\u00a0[latex]\\{\\;z\\large\\;|\\normalsize z&gt;-3\\,\\,\\text{or}\\,\\,z&lt;-8,\\;z\\in\\mathbb{R}\\}[\/latex]\r\n\r\nInterval notation: [latex]x\\in\\left(-\\infty,-8\\right)\\cup\\left(-3,\\infty\\right)[\/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z&lt;-8[\/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x&gt;-3[\/latex] has solutions that continue all the way to the right.\r\n\r\nGraph:<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182846\/image080.jpg\" alt=\"Number line with open circle on negative 8 and an arrow pointed to the left. Also an open circle on negative 3 and an arrow pointed to the right.\" width=\"575\" height=\"53\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nThe following video contains an example of solving a compound inequality involving <em>or\u00a0<\/em><span style=\"font-size: 1rem; text-align: initial;\">and drawing the associated graph.<\/span>\r\n\r\nhttps:\/\/youtu.be\/oRlJ8G7trR8\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.2.3-1.docx\">Transcript-6.2.3-1<\/a>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3921[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Intersections<\/h2>\r\nThe solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the <strong>intersection<\/strong> of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an <i>and<\/i> compound inequality are all the solutions that the two inequalities have in common. This is\u00a0where the two graphs overlap.\r\n\r\nIn this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\u00a0[latex] \\displaystyle 1-4x\\le 21\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,5x+2\\ge22[\/latex]\r\n\r\n&nbsp;\r\n<h4><strong>Solution<\/strong><\/h4>\r\nSolve each inequality for <em>x<\/em>.\u00a0Determine the intersection of the solutions.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,1-4x\\le 21\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,5x+2\\ge 22\\\\\\underline{-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,-2}\\\\\\,\\,\\,\\,\\,\\underline{-4x}\\leq \\underline{20}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{5x}\\,\\,\\,\\,\\,\\,\\,\\ge \\underline{20}\\\\\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge -5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 4\\,\\,\\,\\,\\\\\\\\x\\ge -5\\,\\text{and}\\,\\,x\\ge 4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThe number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\\geq4[\/latex], since\u00a0this is where the two graphs overlap.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064013\/image083.jpg\" alt=\"The inequality x is greater than or equal to negative 5 and the inequality x is greater than or equal to 4 plotted on the number line.\" width=\"575\" height=\"53\" \/>\r\n<h4>Answer<\/h4>\r\nSet-builder notation: [latex] \\{x\\;\\large |\\;\\normalsize x\\ge 4,\\;x\\in\\mathbb{R}\\}[\/latex]\r\n\r\nInterval notation: [latex]x\\in\\left[4,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064014\/image084.jpg\" alt=\"The inequality x is greater than or equal to 4 plotted on the number line.\" width=\"575\" height=\"53\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXample<\/h3>\r\nSolve for <em>x<\/em>: \u00a0[latex] \\displaystyle {5}{x}-{2}\\le{3}\\text{ and }{4}{x}{+7}&gt;{3}[\/latex]\r\n\r\n<strong>Solution:<\/strong>\r\n\r\nSolve each inequality separately.\u00a0Find the overlap between the solutions.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,4x+7&gt;\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\frac{5x}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{5}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{4x}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&gt;\\frac{-4}{4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;-1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x&gt;-1\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nSet-builder notation: [latex]\\{\\;x\\large\\; |\\normalsize\\;-1\\le{x}\\le{1},\\;x\\in\\mathbb{R}\\}[\/latex]\r\n\r\nInterval notation: [latex]x\\in\\left(-1,1\\right][\/latex]\r\n\r\nGraph:<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06231720\/image085.jpg\" alt=\"The inequality negative one less than x less than or equal to one, plotted on the number line.\" width=\"575\" height=\"53\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nNotice that the two inequalities [latex]x\\gt -1[\/latex] and [latex]x\\leq 1[\/latex] can be combined and written as [latex]-1\\lt x\\leq 1[\/latex]. This is best seen on the graph where the solution set lies between [latex]-1[\/latex] and [latex]1[\/latex]. In other words, [latex]x[\/latex] is trapped between\u00a0[latex]-1[\/latex] and [latex]1[\/latex]. All intersection inequalities\u00a0[latex]x\\gt a[\/latex] and [latex]x\\lt b[\/latex] can be written in the form\u00a0[latex]a\\lt x\\lt b[\/latex] provided [latex]a\\lt b[\/latex]. If [latex]a\\gt b[\/latex] there is no graphical overlap which means there is no solution.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3920[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Intersections in the form [latex]a&lt;x&lt;b[\/latex]<\/h2>\r\nA compound inequality that is an intersection can be written in the form [latex]a\\lt x\\lt b[\/latex] rather than [latex]x\\gt a \\text{ and }x\\lt b[\/latex] provided [latex]a\\lt b[\/latex]. If [latex]a\\gt b[\/latex] there is no solution. Rather than splitting a compound inequality in the form of\u00a0\u00a0[latex]a&lt;x&lt;b[\/latex]\u00a0into two inequalities [latex]x\\gt a[\/latex]<i> and <\/i>[latex]x\\lt b[\/latex], we can more quickly solve the inequality by applying the properties of inequality to all three segments of the compound inequality.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>. [latex]3\\lt2x+3\\leq 7[\/latex]\r\n\r\n&nbsp;\r\n<h4><strong>Solution<\/strong><\/h4>\r\nIsolate the variable by subtracting [latex]3[\/latex] from all [latex]3[\/latex] parts of the inequality, then dividing each part by \u00a0[latex]2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,3\\,\\,\\lt\\,\\,2x+3\\,\\,\\leq \\,\\,\\,\\,7\\\\\\underline{\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-3}\\,\\\\\\,\\,\\,\\,\\,\\underline{\\,0\\,}\\,\\,\\lt\\,\\,\\,\\,\\underline{2x}\\,\\,\\,\\,\\,\\,\\,\\,\\leq\\,\\,\\,\\underline{\\,4\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\lt x\\leq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nSet-builder notation: [latex]\\{\\;x\\;\\large |\\normalsize\\; 0\\lt{x}\\le 2,\\;x\\in\\mathbb{R}\\}[\/latex]\r\n\r\nIntervalotation n: [latex]x\\in\\left(0,2\\right][\/latex]\r\n\r\nGraph:\r\n\r\n<img class=\"aligncenter wp-image-3962\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10235848\/Screen-Shot-2016-05-10-at-4.58.30-PM-300x77.png\" alt=\"Open dot on zero, closed dot on 2, and line through all numbers between zero and two.\" width=\"366\" height=\"94\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nThe video below shows another example of how to solve an inequality in the form [latex]a&lt;x&lt;b[\/latex].\r\n\r\nhttps:\/\/youtu.be\/UU_KJI59_08\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.2.3-2.docx\">Transcript-6.2.3-2<\/a>\r\n\r\nThe solution to a compound inequality with <i>and<\/i> is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word <i>and<\/i>:\r\n<table style=\"height: 89px;\" width=\"535\">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"2\">Case 1:<\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Description<\/td>\r\n<td>The solution could be all the values between two endpoints<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Inequalities<\/td>\r\n<td>[latex]x\\le{1}[\/latex] and [latex]x\\gt{-1}[\/latex], or as a bounded inequality: [latex]{-1}\\lt{x}\\le{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Interval<\/td>\r\n<td>[latex]\\left(-1,1\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Graphs<\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064043\/image089.jpg\" alt=\"x less than or equal to one, and x greater than negative one, plotted on the number line.\" width=\"575\" height=\"53\" \/>\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064015\/image085.jpg\" alt=\"Negative one less than x less than or equal to one plotted on the number line.\" width=\"575\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<th colspan=\"2\">Case 2:<\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Description<\/td>\r\n<td>The solution could begin at a point on the number line and extend in one direction.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Inequalities<\/td>\r\n<td>[latex]x\\gt3[\/latex] and [latex]x\\ge4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Interval<\/td>\r\n<td>[latex]\\left[4,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">Graphs<\/td>\r\n<td>\u00a0<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064045\/image090.jpg\" alt=\"x greater than negative three, and x greater than or equal to four plotted on the number line.\" width=\"575\" height=\"53\" \/>\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064014\/image084.jpg\" alt=\"Number line. Closed circle on 4 and arrow through all numbers greater than 4. The arrow represents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<th colspan=\"2\">Case 3:<\/th>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Description<\/td>\r\n<td>In cases where there is no overlap between the two inequalities, there is no solution to the compound inequality<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Inequalities<\/td>\r\n<td>[latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Intervals<\/td>\r\n<td>[latex]\\left(-\\infty,-3\\right)[\/latex] and [latex]\\left(3,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td scope=\"row\">\u00a0Graph<\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064046\/image091.jpg\" alt=\"x less then negative three, x greater than three plotted on the number line.\" width=\"575\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>.\u00a0[latex]x+2&gt;5[\/latex] and [latex]x+4&lt;5[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nSolve each inequality separately.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}x+2&gt;5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,x+4&lt;5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,&gt;\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,&lt;\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;3\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x&lt;1\\end{array}[\/latex]<\/p>\r\nFind the overlap between the solutions.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064048\/image092.jpg\" alt=\"x less than one, x greater than three, plotted on the number line.\" width=\"575\" height=\"53\" \/>\r\n<h4>Answer<\/h4>\r\nThere is no overlap between [latex] \\displaystyle x&gt;3[\/latex] and [latex]x&lt;1[\/latex], so there is no solution.\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nA compound inequality is a statement of two inequality statements linked together either by the word <i>or<\/i> or by the word <i>and<\/i>. Sometimes, an <i>and<\/i> compound inequality is shown symbolically, like\u00a0[latex]a&lt;x&lt;b[\/latex], and does not even need the word <i>and<\/i>. Because compound inequalities represent either a union or intersection of the individual inequalities, graphing them on a number line can be a helpful way to see or check a solution. Compound inequalities can be manipulated and solved in much the same way any inequality is solved, by paying attention to the properties of inequalities and the rules for solving them.\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Solve compound inequalities with OR (Unions) &#8211; express solutions graphically, in set-builder notation and with interval notation<\/li>\n<li>Solve compound inequalities with AND (Intersections) &#8211; express solutions graphically, in set-builder notation and with interval notation<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>Key words<\/h1>\n<ul>\n<li><strong>Compound Inequality<\/strong>: two inequalities joined as a union or an intersection<\/li>\n<li><strong>Union<\/strong>: values exist in either set. Key word is &#8220;or&#8221;.<\/li>\n<li><strong>Intersection<\/strong>: values exist in both sets. Key word is &#8220;and&#8221;.<\/li>\n<\/ul>\n<\/div>\n<h2>Compound Inequalities<\/h2>\n<p>A <em><strong>compound inequality<\/strong><\/em> is two inequalities joined together with the word &#8220;or&#8221; or the word &#8220;and&#8221;. When the two inequalities are joined with &#8220;or&#8221;, the solution set must satisfy <em>either<\/em> equation. For example, [latex]x\\lt 2 \\text{ or } x\\ge 5[\/latex] tells us that the [latex]x[\/latex]-values must be either less than [latex]2[\/latex] or greater than or equal to [latex]5[\/latex]. On the other hand, [latex]x\\ge 2\\text{ and }x\\lt 5[\/latex] tells us that\u00a0the [latex]x[\/latex]-values must be greater than or equal to [latex]2[\/latex] and also less than [latex]5[\/latex]. When &#8220;or&#8221; is used, the solution set is a <em><strong>union<\/strong><\/em> of the solution sets of both inequalities. When &#8220;and&#8221; is used, the solution set is an <em><strong>intersection<\/strong><\/em>\u00a0of the solution sets of both inequalities.<\/p>\n<h2>Unions<\/h2>\n<p>The solution of a compound inequality that consists of two inequalities joined with the word <em>or<\/em> is the <strong>union<\/strong> of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.<\/p>\n<p>The following example shows how to solve a one-step inequality in the <em>or<\/em> form. Note how each inequality is treated independently until the end, where the solution is described in terms of both inequalities. We will use the same properties to solve compound inequalities that we used to solve single inequalities.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]x[\/latex]. \u00a0[latex]3x\u20131<8[\/latex] <em>or<\/em> [latex]x\u20135>0[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Solve each inequality by isolating the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x-5>0\\,\\,\\,\\,\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3x-1<8\\,\\,\\\\\\underline{\\,\\,\\,+5\\,\\,+5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,\\,+1}\\\\x\\,\\,>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{3x}\\,\\,\\,<\\underline{9}\\\\{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{3}\\\\x<3\\,\\,\\,\\\\x>5\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,x<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Set-builder notation: [latex]\\{\\;x\\large\\;|\\;\\normalsize x\\lt 3\\text{ or }x\\gt 5,\\;y\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>Interval notation: [latex]\\left(-\\infty, 3\\right)\\cup\\left(5,\\infty\\right)[\/latex]<\/p>\n<p>The solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first, before writing the solution using interval notation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182844\/image078.jpg\" alt=\"Number line. Open red circle on 3 and red highlight through all numbers less than 3. Open blue circle on 5 and blue highlight on all numbers greater than 5.\" width=\"575\" height=\"53\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for [latex]y[\/latex].\u00a0\u00a0[latex]2y+7\\lt13[\/latex] <em>or<\/em> [latex]\u22123y\u20132\\lt10[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Solve each inequality separately.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}2y+7<13\\,\\,\\,\\,\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-3y-2\\lt 10\\\\\\underline{\\,\\,\\,-7\\,\\,-7}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+2\\,\\,\\,+2}\\\\\\underline{2y}<\\underline{6}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-3y}<\\underline{12}\\\\{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-3}\\\\y<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\gt -4\\\\y<3\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,y\\gt -4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>The inequality sign is reversed with division by a negative number.<\/p>\n<p>Since [latex]y[\/latex] could be less than\u00a0[latex]3[\/latex] or greater than [latex]\u22124,\\;y[\/latex]\u00a0could be any real number. Graphing the inequality helps with this interpretation.<\/p>\n<p>Graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1888\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/09\/28004312\/Union.png\" alt=\"Line graph\" width=\"524\" height=\"117\" \/><\/p>\n<p>The individual graphs show empty dots at [latex]y=3[\/latex] and [latex]y=-4[\/latex], but these graphs combine to all real numbers.<\/p>\n<p>Set-builder notation: [latex]\\{\\;y\\large\\;|\\;\\normalsize \\;y\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>Interval notation: [latex]x\\in\\left(-\\infty,\\infty\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the last example, the final answer included solutions whose intervals overlapped. This caused the answer to include all numbers on the number line. In words, we call this solution &#8220;all real numbers&#8221;.\u00a0Any real number will produce a true statement for either\u00a0[latex]y<3\\text{ or }y\\gt -4[\/latex] when it is substituted for <em>y<\/em>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for [latex]z[\/latex].<\/p>\n<p>[latex]5z\u20133\\gt\u221218[\/latex] <em>or<\/em> [latex]\u22122z\u20131\\gt15[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Solve each inequality separately.\u00a0Combine the solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}5z-3>18\\,\\,\\,\\,\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2z-1>15\\\\\\underline{\\,\\,\\,+3\\,\\,\\,+3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,\\,\\,+1}\\\\\\underline{5z}>\\underline{-15}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2z}>\\underline{16}\\\\{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{-2}\\\\z>-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z<-8\\\\z>-3\\,\\,\\,\\,\\textit{or}\\,\\,\\,\\,z<-8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Set-builder notation:\u00a0[latex]\\{\\;z\\large\\;|\\normalsize z>-3\\,\\,\\text{or}\\,\\,z<-8,\\;z\\in\\mathbb{R}\\}[\/latex]\n\nInterval notation: [latex]x\\in\\left(-\\infty,-8\\right)\\cup\\left(-3,\\infty\\right)[\/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z<-8[\/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[\/latex] has solutions that continue all the way to the right.<\/p>\n<p>Graph:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182846\/image080.jpg\" alt=\"Number line with open circle on negative 8 and an arrow pointed to the left. Also an open circle on negative 3 and an arrow pointed to the right.\" width=\"575\" height=\"53\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The following video contains an example of solving a compound inequality involving <em>or\u00a0<\/em><span style=\"font-size: 1rem; text-align: initial;\">and drawing the associated graph.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solve a Compound Inequality Involving OR (Union)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/oRlJ8G7trR8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.2.3-1.docx\">Transcript-6.2.3-1<\/a><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3921\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3921&theme=oea&iframe_resize_id=ohm3921&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Intersections<\/h2>\n<p>The solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the <strong>intersection<\/strong> of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an <i>and<\/i> compound inequality are all the solutions that the two inequalities have in common. This is\u00a0where the two graphs overlap.<\/p>\n<p>In this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.\u00a0[latex]\\displaystyle 1-4x\\le 21\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,5x+2\\ge22[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Solve each inequality for <em>x<\/em>.\u00a0Determine the intersection of the solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,1-4x\\le 21\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,5x+2\\ge 22\\\\\\underline{-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,-2}\\\\\\,\\,\\,\\,\\,\\underline{-4x}\\leq \\underline{20}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{5x}\\,\\,\\,\\,\\,\\,\\,\\ge \\underline{20}\\\\\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge -5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 4\\,\\,\\,\\,\\\\\\\\x\\ge -5\\,\\text{and}\\,\\,x\\ge 4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>The number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\\geq4[\/latex], since\u00a0this is where the two graphs overlap.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064013\/image083.jpg\" alt=\"The inequality x is greater than or equal to negative 5 and the inequality x is greater than or equal to 4 plotted on the number line.\" width=\"575\" height=\"53\" \/><\/p>\n<h4>Answer<\/h4>\n<p>Set-builder notation: [latex]\\{x\\;\\large |\\;\\normalsize x\\ge 4,\\;x\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>Interval notation: [latex]x\\in\\left[4,\\infty\\right)[\/latex]<\/p>\n<p>Graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064014\/image084.jpg\" alt=\"The inequality x is greater than or equal to 4 plotted on the number line.\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXample<\/h3>\n<p>Solve for <em>x<\/em>: \u00a0[latex]\\displaystyle {5}{x}-{2}\\le{3}\\text{ and }{4}{x}{+7}>{3}[\/latex]<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Solve each inequality separately.\u00a0Find the overlap between the solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,4x+7>\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\frac{5x}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{5}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{4x}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>\\frac{-4}{4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>-1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x>-1\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Set-builder notation: [latex]\\{\\;x\\large\\; |\\normalsize\\;-1\\le{x}\\le{1},\\;x\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>Interval notation: [latex]x\\in\\left(-1,1\\right][\/latex]<\/p>\n<p>Graph:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06231720\/image085.jpg\" alt=\"The inequality negative one less than x less than or equal to one, plotted on the number line.\" width=\"575\" height=\"53\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Notice that the two inequalities [latex]x\\gt -1[\/latex] and [latex]x\\leq 1[\/latex] can be combined and written as [latex]-1\\lt x\\leq 1[\/latex]. This is best seen on the graph where the solution set lies between [latex]-1[\/latex] and [latex]1[\/latex]. In other words, [latex]x[\/latex] is trapped between\u00a0[latex]-1[\/latex] and [latex]1[\/latex]. All intersection inequalities\u00a0[latex]x\\gt a[\/latex] and [latex]x\\lt b[\/latex] can be written in the form\u00a0[latex]a\\lt x\\lt b[\/latex] provided [latex]a\\lt b[\/latex]. If [latex]a\\gt b[\/latex] there is no graphical overlap which means there is no solution.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3920\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3920&theme=oea&iframe_resize_id=ohm3920&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Intersections in the form [latex]a<x<b[\/latex]<\/h2>\n<p>A compound inequality that is an intersection can be written in the form [latex]a\\lt x\\lt b[\/latex] rather than [latex]x\\gt a \\text{ and }x\\lt b[\/latex] provided [latex]a\\lt b[\/latex]. If [latex]a\\gt b[\/latex] there is no solution. Rather than splitting a compound inequality in the form of\u00a0\u00a0[latex]a<x<b[\/latex]\u00a0into two inequalities [latex]x\\gt a[\/latex]<i> and <\/i>[latex]x\\lt b[\/latex], we can more quickly solve the inequality by applying the properties of inequality to all three segments of the compound inequality.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>. [latex]3\\lt2x+3\\leq 7[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Isolate the variable by subtracting [latex]3[\/latex] from all [latex]3[\/latex] parts of the inequality, then dividing each part by \u00a0[latex]2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,3\\,\\,\\lt\\,\\,2x+3\\,\\,\\leq \\,\\,\\,\\,7\\\\\\underline{\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-3}\\,\\\\\\,\\,\\,\\,\\,\\underline{\\,0\\,}\\,\\,\\lt\\,\\,\\,\\,\\underline{2x}\\,\\,\\,\\,\\,\\,\\,\\,\\leq\\,\\,\\,\\underline{\\,4\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\lt x\\leq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Set-builder notation: [latex]\\{\\;x\\;\\large |\\normalsize\\; 0\\lt{x}\\le 2,\\;x\\in\\mathbb{R}\\}[\/latex]<\/p>\n<p>Intervalotation n: [latex]x\\in\\left(0,2\\right][\/latex]<\/p>\n<p>Graph:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3962\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10235848\/Screen-Shot-2016-05-10-at-4.58.30-PM-300x77.png\" alt=\"Open dot on zero, closed dot on 2, and line through all numbers between zero and two.\" width=\"366\" height=\"94\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The video below shows another example of how to solve an inequality in the form [latex]a<x<b[\/latex].\n\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Solve a Compound Inequality Involving AND (Intersection)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UU_KJI59_08?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.2.3-2.docx\">Transcript-6.2.3-2<\/a><\/p>\n<p>The solution to a compound inequality with <i>and<\/i> is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word <i>and<\/i>:<\/p>\n<table style=\"height: 89px; width: 535px;\">\n<tbody>\n<tr>\n<th colspan=\"2\">Case 1:<\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">Description<\/td>\n<td>The solution could be all the values between two endpoints<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Inequalities<\/td>\n<td>[latex]x\\le{1}[\/latex] and [latex]x\\gt{-1}[\/latex], or as a bounded inequality: [latex]{-1}\\lt{x}\\le{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Interval<\/td>\n<td>[latex]\\left(-1,1\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Graphs<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064043\/image089.jpg\" alt=\"x less than or equal to one, and x greater than negative one, plotted on the number line.\" width=\"575\" height=\"53\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064015\/image085.jpg\" alt=\"Negative one less than x less than or equal to one plotted on the number line.\" width=\"575\" height=\"53\" \/><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2\">Case 2:<\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">Description<\/td>\n<td>The solution could begin at a point on the number line and extend in one direction.<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Inequalities<\/td>\n<td>[latex]x\\gt3[\/latex] and [latex]x\\ge4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Interval<\/td>\n<td>[latex]\\left[4,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">Graphs<\/td>\n<td>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064045\/image090.jpg\" alt=\"x greater than negative three, and x greater than or equal to four plotted on the number line.\" width=\"575\" height=\"53\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064014\/image084.jpg\" alt=\"Number line. Closed circle on 4 and arrow through all numbers greater than 4. The arrow represents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/td>\n<\/tr>\n<tr>\n<th colspan=\"2\">Case 3:<\/th>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Description<\/td>\n<td>In cases where there is no overlap between the two inequalities, there is no solution to the compound inequality<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Inequalities<\/td>\n<td>[latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Intervals<\/td>\n<td>[latex]\\left(-\\infty,-3\\right)[\/latex] and [latex]\\left(3,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td scope=\"row\">\u00a0Graph<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064046\/image091.jpg\" alt=\"x less then negative three, x greater than three plotted on the number line.\" width=\"575\" height=\"53\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>.\u00a0[latex]x+2>5[\/latex] and [latex]x+4<5[\/latex]\n\n\n<h4><strong>Solution<\/strong><\/h4>\n<p>Solve each inequality separately.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}x+2>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,x+4<5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,<\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>3\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x<1\\end{array}[\/latex]<\/p>\n<p>Find the overlap between the solutions.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064048\/image092.jpg\" alt=\"x less than one, x greater than three, plotted on the number line.\" width=\"575\" height=\"53\" \/><\/p>\n<h4>Answer<\/h4>\n<p>There is no overlap between [latex]\\displaystyle x>3[\/latex] and [latex]x<1[\/latex], so there is no solution.\n\n<\/div>\n<h2>Summary<\/h2>\n<p>A compound inequality is a statement of two inequality statements linked together either by the word <i>or<\/i> or by the word <i>and<\/i>. Sometimes, an <i>and<\/i> compound inequality is shown symbolically, like\u00a0[latex]a<x<b[\/latex], and does not even need the word <i>and<\/i>. Because compound inequalities represent either a union or intersection of the individual inequalities, graphing them on a number line can be a helpful way to see or check a solution. Compound inequalities can be manipulated and solved in much the same way any inequality is solved, by paying attention to the properties of inequalities and the rules for solving them.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":370291,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2857","chapter","type-chapter","status-publish","hentry"],"part":2842,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2857","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2857\/revisions"}],"predecessor-version":[{"id":3074,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2857\/revisions\/3074"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2842"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2857\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2857"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2857"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2857"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2857"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}