{"id":456,"date":"2016-06-01T20:49:45","date_gmt":"2016-06-01T20:49:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=456"},"modified":"2019-07-24T21:02:40","modified_gmt":"2019-07-24T21:02:40","slug":"read-solve-compound-inequalities-and-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-solve-compound-inequalities-and-2\/","title":{"raw":"Solve Compound Inequalities\u2014AND","rendered":"Solve Compound Inequalities\u2014AND"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Express solutions to inequalities graphically and using interval notation<\/li>\r\n \t<li>Identify solutions for compound inequalities in the form [latex]a&lt;x&lt;b[\/latex] including cases with no solution<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Solve Compound Inequalities in the Form of \"<i>and\"<\/i> and Express the Solution Graphically<\/h2>\r\nThe solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the intersection 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. As we saw in previous sections, 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[reveal-answer q=\"266032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266032\"]\r\n\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\\,\\,\\,\\,\\,\\,\\,\\,\\textit{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\\,\\textit{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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182847\/image083.jpg\" alt=\"Number line. Closed blue circle on negative 5 and blue arrow through all numbers greater than negative 5. This blue arrow is labeled x is greater than or equal to negative 5. Closed red circle on 4 and red arrow through all numbers greater than 4. This red line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/>\r\n\r\nInequality notation: [latex] \\displaystyle x\\ge 4[\/latex]\r\n\r\nInterval notation: [latex]\\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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182849\/image084.jpg\" alt=\"Number line. Closed purple circle (overlapping red and blue circles) on 4 and purple arrow through all numbers greater than 4. Purple line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\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[reveal-answer q=\"784358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"784358\"]\r\n\r\nSolve each inequality separately.\u00a0Find the overlap between the solutions.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\textit{and}\\,\\,\\,\\,\\,\\,\\,4x+7&gt;\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\,\\,\\underline{5x}\\le\\underline{\\,5\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{4x}&gt; \\underline{-4}\\\\\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{4}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;-1\\,\\,\\\\\\\\x\\le1\\,\\textit{and}\\,\\,x&gt;-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nInequality notation: [latex]-1&lt;{x}\\le{1}[\/latex]\r\n\r\nInterval notation: [latex](-1,1][\/latex]\r\n\r\nGraph:<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182704\/image085.jpg\" alt=\"image\" width=\"575\" height=\"53\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Compound Inequalities in the Form [latex]a&lt;x&lt;b[\/latex]<\/h2>\r\nRather than splitting a compound inequality in the form of\u00a0\u00a0[latex]a&lt;x&lt;b[\/latex]\u00a0into two inequalities [latex]x&lt;b[\/latex]<i> and <\/i>[latex]x&gt;a[\/latex], you can more quickly solve the inequality by applying the properties of inequalities 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[reveal-answer q=\"39150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"39150\"]\r\n\r\nIsolate the variable by subtracting 3 from all 3 parts of the inequality then dividing each part by 2.\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\nInequality notation: [latex] \\displaystyle 0\\lt{x}\\le 2[\/latex]\r\n\r\nInterval notation: [latex]\\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\/121\/2016\/06\/01182851\/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[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the video below, you will see another example of how to solve an inequality in the form \u00a0[latex]a&lt;x&lt;b[\/latex].\r\n\r\nhttps:\/\/youtu.be\/UU_KJI59_08\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the compound inequality with variables in all three parts: [latex]3+x&gt;7x - 2&gt;5x - 10[\/latex].\r\n[reveal-answer q=\"343986\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343986\"]\r\n\r\nLet us try the first method: write two inequalities.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3+x&gt; 7x - 2\\hfill &amp; \\textit{and}\\hfill &amp; 7x - 2&gt; 5x - 10\\hfill \\\\ 3&gt; 6x - 2\\hfill &amp; \\hfill &amp; 2x - 2&gt; -10\\hfill \\\\ 5&gt; 6x\\hfill &amp; \\hfill &amp; 2x&gt; -8\\hfill \\\\ \\dfrac{5}{6}&gt; x\\hfill &amp; \\hfill &amp; x&gt; -4\\hfill \\\\ x&lt; \\dfrac{5}{6}\\hfill &amp; \\hfill &amp; -4&lt; x\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is [latex]-4&lt;x&lt;\\dfrac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\dfrac{5}{6}\\normalsize\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right as they appear on a number line.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200410\/CNX_CAT_Figure_02_07_003.jpg\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo solve inequalities like [latex]a&lt;x&lt;b[\/latex], use the addition and multiplication properties of inequality to solve the inequality for <i>x<\/i>. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.\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: 535px;\">\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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182852\/image089.jpg\" alt=\"Number line. Open blue circle on negative 1 and blue arrow through all numbers greater than negative 1. The blue arrow represents x is greater than negative 1. Closed red circle on 1 and red arrow through all numbers less than 1. Red arrow written x is less than or equal to 1.\" width=\"575\" height=\"53\" \/><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182854\/image085.jpg\" alt=\"Number line. Open blue circle on negative 1. Closed red circle on 1. Overlapping red and blue lines between negative 1 and 1 that represents negative 1 is less than x is less than or equal to 1.\" 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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182856\/image090.jpg\" alt=\"Number line. Blue open circle on negative 3 and blue arrow through all numbers greater than negative 3. Blue arrow represents x is greater than negative three. Closed red circle on 4 and red arrow through all numbers greater than 4. The red arrow respresents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182849\/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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182842\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" 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] <em>and<\/em> [latex]x+4&lt;5[\/latex]\r\n\r\n[reveal-answer q=\"336256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"336256\"]\r\n\r\nSolve each inequality separately.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}x+2&gt;5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\textit{and}\\,\\,\\,\\,\\,\\,\\,x+4&lt;5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,x&gt;\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&lt;\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x&gt;3\\,\\,\\,\\,\\textit{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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182858\/image092.jpg\" alt=\"Number line. Red open circle is on 1 and red arrow through all numbers less than 1. Red arrow is labeled x is less than 1. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/>\r\n\r\nThere is no overlap between [latex] \\displaystyle x&gt;3[\/latex] and [latex]x&lt;1[\/latex], so there is no solution.[\/hidden-answer]\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<h3><\/h3>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Express solutions to inequalities graphically and using interval notation<\/li>\n<li>Identify solutions for compound inequalities in the form [latex]a<x<b[\/latex] including cases with no solution<\/li>\n<\/ul>\n<\/div>\n<h2>Solve Compound Inequalities in the Form of &#8220;<i>and&#8221;<\/i> and Express the Solution Graphically<\/h2>\n<p>The solution of a compound inequality that consists of two inequalities joined with the word<i> and <\/i>is the intersection 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. As we saw in previous sections, 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266032\">Show Solution<\/span><\/p>\n<div id=\"q266032\" class=\"hidden-answer\" style=\"display: none\">\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\\,\\,\\,\\,\\,\\,\\,\\,\\textit{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\\,\\textit{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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182847\/image083.jpg\" alt=\"Number line. Closed blue circle on negative 5 and blue arrow through all numbers greater than negative 5. This blue arrow is labeled x is greater than or equal to negative 5. Closed red circle on 4 and red arrow through all numbers greater than 4. This red line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Inequality notation: [latex]\\displaystyle x\\ge 4[\/latex]<\/p>\n<p>Interval notation: [latex]\\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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182849\/image084.jpg\" alt=\"Number line. Closed purple circle (overlapping red and blue circles) on 4 and purple arrow through all numbers greater than 4. Purple line is labeled x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q784358\">Show Solution<\/span><\/p>\n<div id=\"q784358\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality separately.\u00a0Find the overlap between the solutions.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\textit{and}\\,\\,\\,\\,\\,\\,\\,4x+7>\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\,\\,\\underline{5x}\\le\\underline{\\,5\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{4x}> \\underline{-4}\\\\\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{4}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>-1\\,\\,\\\\\\\\x\\le1\\,\\textit{and}\\,\\,x>-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Inequality notation: [latex]-1<{x}\\le{1}[\/latex]\n\nInterval notation: [latex](-1,1][\/latex]\n\nGraph:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182704\/image085.jpg\" alt=\"image\" width=\"575\" height=\"53\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Compound Inequalities in the Form [latex]a<x<b[\/latex]<\/h2>\n<p>Rather than splitting a compound inequality in the form of\u00a0\u00a0[latex]a<x<b[\/latex]\u00a0into two inequalities [latex]x<b[\/latex]<i> and <\/i>[latex]x>a[\/latex], you can more quickly solve the inequality by applying the properties of inequalities 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q39150\">Show Solution<\/span><\/p>\n<div id=\"q39150\" class=\"hidden-answer\" style=\"display: none\">\n<p>Isolate the variable by subtracting 3 from all 3 parts of the inequality then dividing each part by 2.<\/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<p>Inequality notation: [latex]\\displaystyle 0\\lt{x}\\le 2[\/latex]<\/p>\n<p>Interval notation: [latex]\\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\/121\/2016\/06\/01182851\/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<\/div>\n<\/div>\n<\/div>\n<p>In the video below, you will see another example of how to solve an inequality in the form \u00a0[latex]a<x<b[\/latex].\n\n<iframe loading=\"lazy\" id=\"oembed-1\" 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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q343986\">Show Solution<\/span><\/p>\n<div id=\"q343986\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let us try the first method: write two inequalities.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3+x> 7x - 2\\hfill & \\textit{and}\\hfill & 7x - 2> 5x - 10\\hfill \\\\ 3> 6x - 2\\hfill & \\hfill & 2x - 2> -10\\hfill \\\\ 5> 6x\\hfill & \\hfill & 2x> -8\\hfill \\\\ \\dfrac{5}{6}> x\\hfill & \\hfill & x> -4\\hfill \\\\ x< \\dfrac{5}{6}\\hfill & \\hfill & -4< x\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is [latex]-4<x<\\dfrac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\dfrac{5}{6}\\normalsize\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right as they appear on a number line.\n\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200410\/CNX_CAT_Figure_02_07_003.jpg\" alt=\"A number line with the points -4 and 5\/6 labeled. Dots appear at these points and a line connects these two dots.\" width=\"487\" height=\"69\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To solve inequalities like [latex]a<x<b[\/latex], use the addition and multiplication properties of inequality to solve the inequality for <i>x<\/i>. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.<\/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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182852\/image089.jpg\" alt=\"Number line. Open blue circle on negative 1 and blue arrow through all numbers greater than negative 1. The blue arrow represents x is greater than negative 1. Closed red circle on 1 and red arrow through all numbers less than 1. Red arrow written x is less than or equal to 1.\" width=\"575\" height=\"53\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182854\/image085.jpg\" alt=\"Number line. Open blue circle on negative 1. Closed red circle on 1. Overlapping red and blue lines between negative 1 and 1 that represents negative 1 is less than x is less than or equal to 1.\" 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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182856\/image090.jpg\" alt=\"Number line. Blue open circle on negative 3 and blue arrow through all numbers greater than negative 3. Blue arrow represents x is greater than negative three. Closed red circle on 4 and red arrow through all numbers greater than 4. The red arrow respresents x is greater than or equal to 4.\" width=\"575\" height=\"53\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182849\/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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182842\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" 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] <em>and<\/em> [latex]x+4<5[\/latex]\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q336256\">Show Solution<\/span><\/p>\n<div id=\"q336256\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve each inequality separately.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}x+2>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\textit{and}\\,\\,\\,\\,\\,\\,\\,x+4<5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,x>\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x<\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>3\\,\\,\\,\\,\\textit{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\/wp-content\/uploads\/sites\/121\/2016\/06\/01182858\/image092.jpg\" alt=\"Number line. Red open circle is on 1 and red arrow through all numbers less than 1. Red arrow is labeled x is less than 1. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/><\/p>\n<p>There is no overlap between [latex]\\displaystyle x>3[\/latex] and [latex]x<1[\/latex], so there is no solution.<\/div>\n<\/div>\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<h3><\/h3>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-456\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Solve a Compound Inequality Involving AND (Intersection). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/UU_KJI59_08\">https:\/\/youtu.be\/UU_KJI59_08<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li><strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Solve a Compound Inequality Involving AND (Intersection)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/UU_KJI59_08\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-456","chapter","type-chapter","status-publish","hentry"],"part":1709,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/456\/revisions"}],"predecessor-version":[{"id":5375,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/456\/revisions\/5375"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/parts\/1709"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/456\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/media?parent=456"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=456"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/contributor?post=456"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/wp-json\/wp\/v2\/license?post=456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}