{"id":1768,"date":"2023-10-12T00:32:07","date_gmt":"2023-10-12T00:32:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-linear-inequalities-and-absolute-value-inequalities\/"},"modified":"2025-10-31T20:04:28","modified_gmt":"2025-10-31T20:04:28","slug":"introduction-linear-inequalities-and-absolute-value-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-linear-inequalities-and-absolute-value-inequalities\/","title":{"raw":"Linear and Absolute Value Inequalities","rendered":"Linear and Absolute Value Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use interval notation to express inequalities.<\/li>\r\n \t<li>Use properties of inequalities.<\/li>\r\n \t<li>Solve compound inequalities.<\/li>\r\n \t<li>Solve absolute value inequalities.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIt is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.\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\/25200407\/CNX_CAT_Figure_02_07_001N.jpg\" alt=\"Several red winner\u2019s ribbons lie on a white table.\" width=\"488\" height=\"325\" \/>\r\n<h2>Writing and Manipulating Inequalities<\/h2>\r\nIndicating the solution to an inequality such as [latex]x\\ge 4[\/latex] can be achieved in several ways.\r\n\r\nWe can use a number line as shown below.\u00a0The blue ray begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225859\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/>\r\n\r\nWe can use <strong>set-builder notation<\/strong>: [latex]\\{x|x\\ge 4\\}[\/latex], which translates to \"all real numbers <em>x <\/em>such that <em>x <\/em>is greater than or equal to 4.\" Notice that braces are used to indicate a set.\r\n\r\nThe third method is <strong>interval notation<\/strong>, where solution sets are indicated with parentheses or brackets. The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This is perhaps the most useful method as it applies to concepts studied later in this course and to other higher-level math courses.\r\n\r\nThe main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \"equaled.\" A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty ,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities.\r\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th>Set Indicated<\/th>\r\n<th>Set-Builder Notation<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, but not including <em>a <\/em>or <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a&lt;x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|x&gt;a\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|x\\ge a\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|a\\le x\\lt b\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a&lt;x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a\\le x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x&lt;a\\text{ and }x&gt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers<\/td>\r\n<td>[latex]\\{x|x\\text{ is all real numbers}\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a Number<\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].\r\n\r\n[reveal-answer q=\"143041\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"143041\"]\r\n\r\nUse a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse interval notation to indicate all real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].\r\n\r\n[reveal-answer q=\"814810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814810\"]\r\n\r\n[latex]\\left[-3,5\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=58&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92604&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to <em>a <\/em>or Greater Than or Equal to <em>b<\/em><\/h3>\r\nWrite the interval expressing all real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].\r\n\r\n[reveal-answer q=\"797079\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797079\"]\r\n\r\nWe have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\\infty [\/latex] and ends at [latex]-1[\/latex], which is written as [latex]\\left(-\\infty ,-1\\right][\/latex].\r\n\r\nThe second interval must show all real numbers greater than or equal to [latex]1[\/latex], which is written as [latex]\\left[1,\\infty \\right)[\/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\\cup [\/latex], between the two intervals.\r\n<div style=\"text-align: center;\">[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nExpress all real numbers less than [latex]-2[\/latex] or greater than or equal to 3 in interval notation.\r\n\r\n[reveal-answer q=\"729196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"729196\"]\r\n\r\n[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2748&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Using the Properties of Inequalities<\/h3>\r\nWhen we work with inequalities, we can usually treat them similarly to but not exactly as we treat equations. We can use the <strong>addition property<\/strong> and the <strong>multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number, we must reverse the inequality symbol.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of Inequalities<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\text{Addition Property}\\hfill&amp; \\text{If }a&lt; b,\\text{ then }a+c&lt; b+c.\\hfill \\\\ \\hfill &amp; \\hfill \\\\ \\text{Multiplication Property}\\hfill &amp; \\text{If }a&lt; b\\text{ and }c&gt; 0,\\text{ then }ac&lt; bc.\\hfill \\\\ \\hfill &amp; \\text{If }a&lt; b\\text{ and }c&lt; 0,\\text{ then }ac&gt; bc.\\hfill \\end{array}[\/latex]<\/p>\r\nThese properties also apply to [latex]a\\le b[\/latex], [latex]a&gt;b[\/latex], and [latex]a\\ge b[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Demonstrating the Addition Property<\/h3>\r\nIllustrate the addition property for inequalities by solving each of the following:\r\n<ol>\r\n \t<li>[latex]x - 15&lt;4[\/latex]<\/li>\r\n \t<li>[latex]6\\ge x - 1[\/latex]<\/li>\r\n \t<li>[latex]x+7&gt;9[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"105622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"105622\"]\r\n\r\nThe addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.\r\n\r\n1.\r\n\r\n[latex]\\begin{array}{ll}x - 15&lt;4\\hfill &amp; \\hfill \\\\ x - 15+15&lt;4+15 \\hfill &amp; \\text{Add 15 to both sides.}\\hfill \\\\ x&lt;19\\hfill &amp; \\hfill \\end{array}[\/latex]\r\n\r\n2.\r\n\r\n[latex]\\begin{array}{ll}6\\ge x - 1\\hfill &amp; \\hfill \\\\ 6+1\\ge x - 1+1\\hfill &amp; \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill &amp; \\hfill \\end{array}[\/latex]\r\n\r\n3.\r\n\r\n[latex]\\begin{array}{ll}x+7&gt;9\\hfill &amp; \\hfill \\\\ x+7 - 7&gt;9 - 7\\hfill &amp; \\text{Subtract 7 from both sides}.\\hfill \\\\ x&gt;2\\hfill &amp; \\hfill \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]3x - 2&lt;1[\/latex].\r\n\r\n[reveal-answer q=\"68318\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"68318\"]\r\n\r\n[latex]x&lt;1[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92605&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Demonstrating the Multiplication Property<\/h3>\r\nIllustrate the multiplication property for inequalities by solving each of the following:\r\n<ol>\r\n \t<li>[latex]3x&lt;6[\/latex]<\/li>\r\n \t<li>[latex]-2x - 1\\ge 5[\/latex]<\/li>\r\n \t<li>[latex]5-x&gt;10[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"749552\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"749552\"]\r\n\r\n1.\r\n\r\n[latex]\\begin{array}{l}3x&lt;6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)&lt;\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x&lt;2\\hfill \\end{array}[\/latex]\r\n\r\n2.\r\n\r\n[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill &amp; \\hfill \\\\ -2x\\ge 6\\hfill &amp; \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill &amp; \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill &amp; \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]\r\n\r\n3.\r\n\r\n[latex]\\begin{array}{ll}5-x&gt;10\\hfill &amp; \\hfill \\\\ -x&gt;5\\hfill &amp; \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)&gt;\\left(5\\right)\\left(-1\\right)\\hfill &amp; \\text{Multiply by }-1.\\hfill \\\\ x&lt;-5\\hfill &amp; \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]4x+7\\ge 2x - 3[\/latex].\r\n\r\n[reveal-answer q=\"32307\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"32307\"]\r\n\r\n[latex]x\\ge -5[\/latex][\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92606&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Solving Inequalities in One Variable Algebraically<\/h3>\r\nAs the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Inequality Algebraically<\/h3>\r\nSolve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].\r\n\r\n[reveal-answer q=\"453286\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453286\"]\r\n\r\nSolving this inequality is similar to solving an equation up until the last step.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill &amp; \\hfill \\\\ 13 - 17x\\ge -4\\hfill &amp; \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill &amp; \\text{Isolate the variable term}.\\hfill \\\\ x\\le 1\\hfill &amp; \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality and write the answer using interval notation: [latex]-x+4&lt;\\frac{1}{2}x+1[\/latex].\r\n\r\n[reveal-answer q=\"703883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703883\"]\r\n\r\n[latex]\\left(2,\\infty \\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92607&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Inequality with Fractions<\/h3>\r\nSolve the following inequality and write the answer in interval notation: [latex]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].\r\n\r\n[reveal-answer q=\"37354\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"37354\"]\r\nWe begin solving in the same way we do when solving an equation.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill &amp; \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill &amp; \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill &amp; \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill &amp; \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill &amp; \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is the interval [latex]\\left(-\\infty ,\\frac{15}{34}\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].\r\n\r\n[reveal-answer q=\"524889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"524889\"]\r\n\r\n[latex]\\left[-\\frac{3}{14},\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=72891&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Compound and Absolute Value Inequalities<\/h2>\r\nA <strong>compound inequality<\/strong> includes two inequalities in one statement. A statement such as [latex]4\\lt x\\le 6[\/latex] means [latex]4\\lt x[\/latex] and [latex]x\\le 6[\/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Compound Inequality<\/h3>\r\nSolve the compound inequality: [latex]3\\le 2x+2&lt;6[\/latex].\r\n\r\n[reveal-answer q=\"497940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497940\"]\r\n\r\nThe first method is to write two separate inequalities: [latex]3\\le 2x+2[\/latex] and [latex]2x+2&lt;6[\/latex]. We solve them independently.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3\\le 2x+2\\hfill &amp; \\text{and}\\hfill &amp; 2x+2&lt;6\\hfill \\\\ 1\\le 2x\\hfill &amp; \\hfill &amp; 2x&lt;4\\hfill \\\\ \\frac{1}{2}\\le x\\hfill &amp; \\hfill &amp; x&lt;2\\hfill \\end{array}[\/latex]<\/div>\r\nThen, we can rewrite the solution as a compound inequality, the same way the problem began.\r\n<div style=\"text-align: center;\">[latex]\\frac{1}{2}\\le x&lt;2[\/latex]<\/div>\r\nIn interval notation, the solution is written as [latex]\\left[\\frac{1}{2},2\\right)[\/latex].\r\n\r\nThe second method is to leave the compound inequality intact and perform solving procedures on the three parts at the same time.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}3\\le 2x+2&lt;6\\hfill &amp; \\hfill \\\\ 1\\le 2x&lt;4\\hfill &amp; \\text{Isolate the variable term and subtract 2 from all three parts}.\\hfill \\\\ \\frac{1}{2}\\le x&lt;2\\hfill &amp; \\text{Divide through all three parts by 2}.\\hfill \\end{array}[\/latex]<\/div>\r\nWe get the same solution: [latex]\\left[\\frac{1}{2},2\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the compound inequality [latex]4\\lt 2x - 8\\le 10[\/latex].\r\n\r\n[reveal-answer q=\"265531\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265531\"]\r\n\r\n[latex]6\\lt x\\le 9\\text{ }\\text{ }\\text{or}\\left(6,9\\right][\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92608&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\r\nSolve the compound inequality with variables in all three parts: [latex]3+x&gt;7x - 2&gt;5x - 10[\/latex].\r\n\r\n[reveal-answer q=\"658677\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"658677\"]\r\n\r\nLet's try the first method. Write two inequalities<strong>:<\/strong>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3+x&gt; 7x - 2\\hfill &amp; \\text{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 \\\\ \\frac{5}{6}&gt; x\\hfill &amp; \\hfill &amp; x&gt; -4\\hfill \\\\ x&lt; \\frac{5}{6}\\hfill &amp; \\hfill &amp; -4&lt; x\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is [latex]-4&lt;x&lt;\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\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 class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225901\/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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the compound inequality: [latex]3y&lt;4 - 5y&lt;5+3y[\/latex].\r\n\r\n[reveal-answer q=\"661493\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"661493\"]\r\n\r\n[latex]\\left(-\\frac{1}{8},\\frac{1}{2}\\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92609&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Solving Absolute Value Inequalities<\/h3>\r\nAs we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at [latex]\\left(-x,0\\right)[\/latex] has an absolute value of [latex]x[\/latex] as it is <em>x <\/em>units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.\r\n\r\nAn <strong>absolute value inequality<\/strong> is an equation of the form\r\n<div style=\"text-align: center;\">[latex]|A|\\lt B,|A|\\le B,|A| \\gt B,\\text{or }|A|\\ge B[\/latex],<\/div>\r\nwhere <em>A<\/em>, and sometimes <em>B<\/em>, represents an algebraic expression dependent on a variable <em>x. <\/em>Solving the inequality means finding the set of all [latex]x[\/latex] <em>-<\/em>values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.\r\n\r\nThere are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.\r\n\r\nSuppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of <em>x-<\/em>values such that the distance between [latex]x[\/latex] and 600 is less than 200. We represent the distance between [latex]x[\/latex] and 600 as [latex]|x - 600|[\/latex], and therefore, [latex]|x - 600|\\le 200[\/latex] or\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{c}-200\\le x - 600\\le 200\\\\ -200+600\\le x - 600+600\\le 200+600\\\\ 400\\le x\\le 800\\end{array}[\/latex]<\/div>\r\nThis means our returns would be between $400 and $800.\r\n\r\nTo solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Absolute Value Inequalities<\/h3>\r\nFor an algebraic expression [latex]X[\/latex]<em>\u00a0<\/em>and [latex]k&gt;0[\/latex], an <strong>absolute value inequality<\/strong> is an inequality of the form:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}|X|&lt; k\\text{ which is equivalent to }-k&lt; X&lt; k\\hfill \\text{ or }\\ |X|&gt; k\\text{ which is equivalent to }X&lt; -k\\text{ or }X&gt; k\\hfill \\end{array}[\/latex]<\/div>\r\nThese statements also apply to [latex]|X|\\le k[\/latex] and [latex]|X|\\ge k[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Determining a Number within a Prescribed Distance<\/h3>\r\nDescribe all values [latex]x[\/latex] within a distance of 4 from the number 5.\r\n\r\n[reveal-answer q=\"746497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746497\"]\r\n\r\nWe want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225903\/CNX_Precalc_Figure_01_06_002.jpg\" alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\" width=\"487\" height=\"81\" \/>\r\n\r\nThe distance from [latex]x[\/latex] to 5 can be represented using an absolute value symbol, [latex]|x - 5|[\/latex]. Write the values of [latex]x[\/latex] that satisfy the condition as an absolute value inequality.\r\n<div style=\"text-align: center;\">[latex]|x - 5|\\le 4[\/latex]<\/div>\r\nWe need to write two inequalities as there are always two solutions to an absolute value equation.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}x - 5\\le 4\\hfill &amp; \\text{and}\\hfill &amp; x - 5\\ge -4\\hfill \\\\ x\\le 9\\hfill &amp; \\hfill &amp; x\\ge 1\\hfill \\end{array}[\/latex]<\/div>\r\nIf the solution set is [latex]x\\le 9[\/latex] and [latex]x\\ge 1[\/latex], then the solution set is an interval including all real numbers between and including 1 and 9.\r\n\r\nSo [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]\\left[1,9\\right][\/latex] in interval notation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe all <em>x-<\/em>values within a distance of 3 from the number 2.\r\n\r\n[reveal-answer q=\"7507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"7507\"]\r\n\r\n[latex]|x - 2|\\le 3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Absolute Value Inequality<\/h3>\r\nSolve [latex]|x - 1|\\le 3[\/latex].\r\n\r\n[reveal-answer q=\"4865\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"4865\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}|x - 1|\\le 3\\hfill \\\\ \\hfill \\\\ -3\\le x - 1\\le 3\\hfill \\\\ \\hfill \\\\ -2\\le x\\le 4\\hfill \\\\ \\hfill \\\\ \\left[-2,4\\right]\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\r\nGiven the equation [latex]y=-\\frac{1}{2}|4x - 5|+3[\/latex], determine the <em>x<\/em>-values for which the <em>y<\/em>-values are negative.\r\n\r\n[reveal-answer q=\"624558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624558\"]\r\n\r\nWe are trying to determine where [latex]y&lt;0[\/latex] which is when [latex]-\\frac{1}{2}|4x - 5|+3&lt;0[\/latex]. We begin by isolating the absolute value.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{1}{2}|4x - 5|&lt; -3\\hfill &amp; \\text{Multiply both sides by -2, and reverse the inequality}.\\hfill \\\\ |4x - 5|&gt; 6\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\nNext, we solve [latex]|4x - 5|=6[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}4x - 5=6\\hfill &amp; \\hfill &amp; 4x - 5=-6\\hfill \\\\ 4x=11\\hfill &amp; \\text{or}\\hfill &amp; 4x=-1\\hfill \\\\ x=\\frac{11}{4}\\hfill &amp; \\hfill &amp; x=-\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nNow, we can examine the graph to observe where the <em>y-<\/em>values are negative. We observe where the branches are below the <em>x-<\/em>axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at [latex]x=-\\frac{1}{4}[\/latex] and [latex]x=\\frac{11}{4}[\/latex] and that the graph opens downward.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225906\/CNX_CAT_Figure_02_07_006.jpg\" alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\" width=\"487\" height=\"363\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]-2|k - 4|\\le -6[\/latex].\r\n\r\n[reveal-answer q=\"96760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96760\"]\r\n\r\n[latex]k\\le 1[\/latex] or [latex]k\\ge 7[\/latex]; in interval notation, this would be [latex]\\left(-\\infty ,1\\right]\\cup \\left[7,\\infty \\right)[\/latex].\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200413\/CNX_CAT_Figure_02_07_007.jpg\" alt=\"A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8. The function y = -2|k 4| + 6 is graphed and everything above the function is shaded in.\" \/>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=89935&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15505&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>Interval notation is a method to give the solution set of an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.<\/li>\r\n \t<li>Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.<\/li>\r\n \t<li>Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.<\/li>\r\n \t<li>Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value.<\/li>\r\n \t<li>Absolute value inequality solutions can be verified by graphing. We can check the algebraic solutions by graphing as we cannot depend on a visual for a precise solution.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>\r\n<dl id=\"fs-id1165131990658\" class=\"definition\">\r\n \t<dt><strong>compound inequality<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165131990661\">a problem or a statement that includes two inequalities<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>interval<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">an interval describes a set of numbers where a solution falls<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt><strong>interval notation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134297639\">a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt><strong>linear inequality<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135486042\">similar to a linear equation except that the solutions will include an interval of numbers<\/dd>\r\n<\/dl>\r\n&nbsp;\r\n<h3 style=\"text-align: center;\"><\/h3>\r\n<\/dt>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use interval notation to express inequalities.<\/li>\n<li>Use properties of inequalities.<\/li>\n<li>Solve compound inequalities.<\/li>\n<li>Solve absolute value inequalities.<\/li>\n<\/ul>\n<\/div>\n<p>It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.<\/p>\n<p><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\/25200407\/CNX_CAT_Figure_02_07_001N.jpg\" alt=\"Several red winner\u2019s ribbons lie on a white table.\" width=\"488\" height=\"325\" \/><\/p>\n<h2>Writing and Manipulating Inequalities<\/h2>\n<p>Indicating the solution to an inequality such as [latex]x\\ge 4[\/latex] can be achieved in several ways.<\/p>\n<p>We can use a number line as shown below.\u00a0The blue ray begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225859\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/><\/p>\n<p>We can use <strong>set-builder notation<\/strong>: [latex]\\{x|x\\ge 4\\}[\/latex], which translates to &#8220;all real numbers <em>x <\/em>such that <em>x <\/em>is greater than or equal to 4.&#8221; Notice that braces are used to indicate a set.<\/p>\n<p>The third method is <strong>interval notation<\/strong>, where solution sets are indicated with parentheses or brackets. The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This is perhaps the most useful method as it applies to concepts studied later in this course and to other higher-level math courses.<\/p>\n<p>The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be &#8220;equaled.&#8221; A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty ,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities.<\/p>\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\n<thead>\n<tr>\n<th>Set Indicated<\/th>\n<th>Set-Builder Notation<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, but not including <em>a <\/em>or <em>b<\/em><\/td>\n<td>[latex]\\{x|a<x<b\\}[\/latex]<\/td>\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td>[latex]\\{x|x>a\\}[\/latex]<\/td>\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td>[latex]\\{x|x<b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\{x|x\\ge a\\}[\/latex]<\/td>\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\{x|x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\{x|a\\le x\\lt b\\}[\/latex]<\/td>\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\{x|a<x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\n<td>[latex]\\{x|a\\le x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td>[latex]\\{x|x<a\\text{ and }x>b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers<\/td>\n<td>[latex]\\{x|x\\text{ is all real numbers}\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to a Number<\/h3>\n<p>Use interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q143041\">Show Solution<\/span><\/p>\n<div id=\"q143041\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use interval notation to indicate all real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814810\">Show Solution<\/span><\/p>\n<div id=\"q814810\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-3,5\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=58&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92604&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to <em>a <\/em>or Greater Than or Equal to <em>b<\/em><\/h3>\n<p>Write the interval expressing all real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797079\">Show Solution<\/span><\/p>\n<div id=\"q797079\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\\infty[\/latex] and ends at [latex]-1[\/latex], which is written as [latex]\\left(-\\infty ,-1\\right][\/latex].<\/p>\n<p>The second interval must show all real numbers greater than or equal to [latex]1[\/latex], which is written as [latex]\\left[1,\\infty \\right)[\/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\\cup[\/latex], between the two intervals.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Express all real numbers less than [latex]-2[\/latex] or greater than or equal to 3 in interval notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729196\">Show Solution<\/span><\/p>\n<div id=\"q729196\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2748&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Using the Properties of Inequalities<\/h3>\n<p>When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equations. We can use the <strong>addition property<\/strong> and the <strong>multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number, we must reverse the inequality symbol.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Inequalities<\/h3>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\text{Addition Property}\\hfill& \\text{If }a< b,\\text{ then }a+c< b+c.\\hfill \\\\ \\hfill & \\hfill \\\\ \\text{Multiplication Property}\\hfill & \\text{If }a< b\\text{ and }c> 0,\\text{ then }ac< bc.\\hfill \\\\ \\hfill & \\text{If }a< b\\text{ and }c< 0,\\text{ then }ac> bc.\\hfill \\end{array}[\/latex]<\/p>\n<p>These properties also apply to [latex]a\\le b[\/latex], [latex]a>b[\/latex], and [latex]a\\ge b[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Demonstrating the Addition Property<\/h3>\n<p>Illustrate the addition property for inequalities by solving each of the following:<\/p>\n<ol>\n<li>[latex]x - 15<4[\/latex]<\/li>\n<li>[latex]6\\ge x - 1[\/latex]<\/li>\n<li>[latex]x+7>9[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q105622\">Show Solution<\/span><\/p>\n<div id=\"q105622\" class=\"hidden-answer\" style=\"display: none\">\n<p>The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.<\/p>\n<p>1.<\/p>\n<p>[latex]\\begin{array}{ll}x - 15<4\\hfill & \\hfill \\\\ x - 15+15<4+15 \\hfill & \\text{Add 15 to both sides.}\\hfill \\\\ x<19\\hfill & \\hfill \\end{array}[\/latex]\n\n2.\n\n[latex]\\begin{array}{ll}6\\ge x - 1\\hfill & \\hfill \\\\ 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill & \\hfill \\end{array}[\/latex]\n\n3.\n\n[latex]\\begin{array}{ll}x+7>9\\hfill & \\hfill \\\\ x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill \\\\ x>2\\hfill & \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]3x - 2<1[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q68318\">Show Solution<\/span><\/p>\n<div id=\"q68318\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x<1[\/latex]\n\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92605&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Demonstrating the Multiplication Property<\/h3>\n<p>Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\n<ol>\n<li>[latex]3x<6[\/latex]<\/li>\n<li>[latex]-2x - 1\\ge 5[\/latex]<\/li>\n<li>[latex]5-x>10[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q749552\">Show Solution<\/span><\/p>\n<div id=\"q749552\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p>[latex]\\begin{array}{l}3x<6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)<\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x<2\\hfill \\end{array}[\/latex]\n\n2.\n\n[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill & \\hfill \\\\ -2x\\ge 6\\hfill & \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill & \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]\n\n3.\n\n[latex]\\begin{array}{ll}5-x>10\\hfill & \\hfill \\\\ -x>5\\hfill & \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)\\hfill & \\text{Multiply by }-1.\\hfill \\\\ x<-5\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]\n\n\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]4x+7\\ge 2x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q32307\">Show Solution<\/span><\/p>\n<div id=\"q32307\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x\\ge -5[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92606&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Solving Inequalities in One Variable Algebraically<\/h3>\n<p>As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inequality Algebraically<\/h3>\n<p>Solve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453286\">Show Solution<\/span><\/p>\n<div id=\"q453286\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solving this inequality is similar to solving an equation up until the last step.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill & \\hfill \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill & \\text{Isolate the variable term}.\\hfill \\\\ x\\le 1\\hfill & \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the inequality and write the answer using interval notation: [latex]-x+4<\\frac{1}{2}x+1[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q703883\">Show Solution<\/span><\/p>\n<div id=\"q703883\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(2,\\infty \\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92607&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inequality with Fractions<\/h3>\n<p>Solve the following inequality and write the answer in interval notation: [latex]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q37354\">Show Solution<\/span><\/p>\n<div id=\"q37354\" class=\"hidden-answer\" style=\"display: none\">\nWe begin solving in the same way we do when solving an equation.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill & \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill & \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill & \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill & \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill & \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is the interval [latex]\\left(-\\infty ,\\frac{15}{34}\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q524889\">Show Solution<\/span><\/p>\n<div id=\"q524889\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-\\frac{3}{14},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom7\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=72891&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Compound and Absolute Value Inequalities<\/h2>\n<p>A <strong>compound inequality<\/strong> includes two inequalities in one statement. A statement such as [latex]4\\lt x\\le 6[\/latex] means [latex]4\\lt x[\/latex] and [latex]x\\le 6[\/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Compound Inequality<\/h3>\n<p>Solve the compound inequality: [latex]3\\le 2x+2<6[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497940\">Show Solution<\/span><\/p>\n<div id=\"q497940\" class=\"hidden-answer\" style=\"display: none\">\n<p>The first method is to write two separate inequalities: [latex]3\\le 2x+2[\/latex] and [latex]2x+2<6[\/latex]. We solve them independently.\n\n\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3\\le 2x+2\\hfill & \\text{and}\\hfill & 2x+2<6\\hfill \\\\ 1\\le 2x\\hfill & \\hfill & 2x<4\\hfill \\\\ \\frac{1}{2}\\le x\\hfill & \\hfill & x<2\\hfill \\end{array}[\/latex]<\/div>\n<p>Then, we can rewrite the solution as a compound inequality, the same way the problem began.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{1}{2}\\le x<2[\/latex]<\/div>\n<p>In interval notation, the solution is written as [latex]\\left[\\frac{1}{2},2\\right)[\/latex].<\/p>\n<p>The second method is to leave the compound inequality intact and perform solving procedures on the three parts at the same time.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}3\\le 2x+2<6\\hfill & \\hfill \\\\ 1\\le 2x<4\\hfill & \\text{Isolate the variable term and subtract 2 from all three parts}.\\hfill \\\\ \\frac{1}{2}\\le x<2\\hfill & \\text{Divide through all three parts by 2}.\\hfill \\end{array}[\/latex]<\/div>\n<p>We get the same solution: [latex]\\left[\\frac{1}{2},2\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the compound inequality [latex]4\\lt 2x - 8\\le 10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265531\">Show Solution<\/span><\/p>\n<div id=\"q265531\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6\\lt x\\le 9\\text{ }\\text{ }\\text{or}\\left(6,9\\right][\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92608&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Compound Inequality with the Variable in All Three Parts<\/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=\"q658677\">Show Solution<\/span><\/p>\n<div id=\"q658677\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s try the first method. Write two inequalities<strong>:<\/strong><\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}3+x> 7x - 2\\hfill & \\text{and}\\hfill & 7x - 2> 5x - 10\\hfill \\\\ 3> 6x - 2\\hfill & \\hfill & 2x - 2> -10\\hfill \\\\ 5> 6x\\hfill & \\hfill & 2x> -8\\hfill \\\\ \\frac{5}{6}> x\\hfill & \\hfill & x> -4\\hfill \\\\ x< \\frac{5}{6}\\hfill & \\hfill & -4< x\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is [latex]-4<x<\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\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\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225901\/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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the compound inequality: [latex]3y<4 - 5y<5+3y[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q661493\">Show Solution<\/span><\/p>\n<div id=\"q661493\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\frac{1}{8},\\frac{1}{2}\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92609&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Solving Absolute Value Inequalities<\/h3>\n<p>As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at [latex]\\left(-x,0\\right)[\/latex] has an absolute value of [latex]x[\/latex] as it is <em>x <\/em>units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.<\/p>\n<p>An <strong>absolute value inequality<\/strong> is an equation of the form<\/p>\n<div style=\"text-align: center;\">[latex]|A|\\lt B,|A|\\le B,|A| \\gt B,\\text{or }|A|\\ge B[\/latex],<\/div>\n<p>where <em>A<\/em>, and sometimes <em>B<\/em>, represents an algebraic expression dependent on a variable <em>x. <\/em>Solving the inequality means finding the set of all [latex]x[\/latex] <em>&#8211;<\/em>values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.<\/p>\n<p>There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.<\/p>\n<p>Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of <em>x-<\/em>values such that the distance between [latex]x[\/latex] and 600 is less than 200. We represent the distance between [latex]x[\/latex] and 600 as [latex]|x - 600|[\/latex], and therefore, [latex]|x - 600|\\le 200[\/latex] or<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{c}-200\\le x - 600\\le 200\\\\ -200+600\\le x - 600+600\\le 200+600\\\\ 400\\le x\\le 800\\end{array}[\/latex]<\/div>\n<p>This means our returns would be between $400 and $800.<\/p>\n<p>To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Absolute Value Inequalities<\/h3>\n<p>For an algebraic expression [latex]X[\/latex]<em>\u00a0<\/em>and [latex]k>0[\/latex], an <strong>absolute value inequality<\/strong> is an inequality of the form:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}|X|< k\\text{ which is equivalent to }-k< X< k\\hfill \\text{ or }\\ |X|> k\\text{ which is equivalent to }X< -k\\text{ or }X> k\\hfill \\end{array}[\/latex]<\/div>\n<p>These statements also apply to [latex]|X|\\le k[\/latex] and [latex]|X|\\ge k[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Determining a Number within a Prescribed Distance<\/h3>\n<p>Describe all values [latex]x[\/latex] within a distance of 4 from the number 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q746497\">Show Solution<\/span><\/p>\n<div id=\"q746497\" class=\"hidden-answer\" style=\"display: none\">\n<p>We want the distance between [latex]x[\/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225903\/CNX_Precalc_Figure_01_06_002.jpg\" alt=\"A number line with one tick mark in the center labeled: 5. The tick marks on either side of the center one are not marked. Arrows extend from the center tick mark to the outer tick marks, both are labeled 4.\" width=\"487\" height=\"81\" \/><\/p>\n<p>The distance from [latex]x[\/latex] to 5 can be represented using an absolute value symbol, [latex]|x - 5|[\/latex]. Write the values of [latex]x[\/latex] that satisfy the condition as an absolute value inequality.<\/p>\n<div style=\"text-align: center;\">[latex]|x - 5|\\le 4[\/latex]<\/div>\n<p>We need to write two inequalities as there are always two solutions to an absolute value equation.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}x - 5\\le 4\\hfill & \\text{and}\\hfill & x - 5\\ge -4\\hfill \\\\ x\\le 9\\hfill & \\hfill & x\\ge 1\\hfill \\end{array}[\/latex]<\/div>\n<p>If the solution set is [latex]x\\le 9[\/latex] and [latex]x\\ge 1[\/latex], then the solution set is an interval including all real numbers between and including 1 and 9.<\/p>\n<p>So [latex]|x - 5|\\le 4[\/latex] is equivalent to [latex]\\left[1,9\\right][\/latex] in interval notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe all <em>x-<\/em>values within a distance of 3 from the number 2.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q7507\">Show Solution<\/span><\/p>\n<div id=\"q7507\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]|x - 2|\\le 3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Absolute Value Inequality<\/h3>\n<p>Solve [latex]|x - 1|\\le 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4865\">Show Solution<\/span><\/p>\n<div id=\"q4865\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}|x - 1|\\le 3\\hfill \\\\ \\hfill \\\\ -3\\le x - 1\\le 3\\hfill \\\\ \\hfill \\\\ -2\\le x\\le 4\\hfill \\\\ \\hfill \\\\ \\left[-2,4\\right]\\hfill \\end{array}[\/latex]<\/p>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Graphical Approach to Solve Absolute Value Inequalities<\/h3>\n<p>Given the equation [latex]y=-\\frac{1}{2}|4x - 5|+3[\/latex], determine the <em>x<\/em>-values for which the <em>y<\/em>-values are negative.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q624558\">Show Solution<\/span><\/p>\n<div id=\"q624558\" class=\"hidden-answer\" style=\"display: none\">\n<p>We are trying to determine where [latex]y<0[\/latex] which is when [latex]-\\frac{1}{2}|4x - 5|+3<0[\/latex]. We begin by isolating the absolute value.\n\n\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{1}{2}|4x - 5|< -3\\hfill & \\text{Multiply both sides by -2, and reverse the inequality}.\\hfill \\\\ |4x - 5|> 6\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p>Next, we solve [latex]|4x - 5|=6[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lll}4x - 5=6\\hfill & \\hfill & 4x - 5=-6\\hfill \\\\ 4x=11\\hfill & \\text{or}\\hfill & 4x=-1\\hfill \\\\ x=\\frac{11}{4}\\hfill & \\hfill & x=-\\frac{1}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p>Now, we can examine the graph to observe where the <em>y-<\/em>values are negative. We observe where the branches are below the <em>x-<\/em>axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at [latex]x=-\\frac{1}{4}[\/latex] and [latex]x=\\frac{11}{4}[\/latex] and that the graph opens downward.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225906\/CNX_CAT_Figure_02_07_006.jpg\" alt=\"A coordinate plan with the x-axis ranging from -5 to 5 and the y-axis ranging from -4 to 4. The function y = -1\/2|4x \u2013 5| + 3 is graphed. An open circle appears at the point -0.25 and an arrow\" width=\"487\" height=\"363\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]-2|k - 4|\\le -6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96760\">Show Solution<\/span><\/p>\n<div id=\"q96760\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]k\\le 1[\/latex] or [latex]k\\ge 7[\/latex]; in interval notation, this would be [latex]\\left(-\\infty ,1\\right]\\cup \\left[7,\\infty \\right)[\/latex].<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200413\/CNX_CAT_Figure_02_07_007.jpg\" alt=\"A coordinate plane with the x-axis ranging from -1 to 9 and the y-axis ranging from -3 to 8. The function y = -2|k 4| + 6 is graphed and everything above the function is shaded in.\" \/><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=89935&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15505&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>Interval notation is a method to give the solution set of an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well.<\/li>\n<li>Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality.<\/li>\n<li>Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities.<\/li>\n<li>Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value.<\/li>\n<li>Absolute value inequality solutions can be verified by graphing. We can check the algebraic solutions by graphing as we cannot depend on a visual for a precise solution.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>compound inequality<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">a problem or a statement that includes two inequalities<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>interval<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">an interval describes a set of numbers where a solution falls<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>interval notation<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>linear inequality<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">similar to a linear equation except that the solutions will include an interval of numbers<\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\n<h3 style=\"text-align: center;\"><\/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-1768\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graphing Inequalities. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/4529rytfef\">https:\/\/www.desmos.com\/calculator\/4529rytfef<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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@5.2<\/li><li>Question ID 92604, 92605, 92606, 92607, 92608, 92609. <strong>Authored by<\/strong>: Michael Jenck. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 72891. <strong>Authored by<\/strong>: Alyson Day. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 58. <strong>Authored by<\/strong>: Lippman, David. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 89935. <strong>Authored by<\/strong>: Krystal Meier. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15505. <strong>Authored by<\/strong>: Tophe Anderson. <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>: IMathAS Community License CC-BY + GPL<\/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":708740,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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