{"id":4902,"date":"2017-12-28T16:54:14","date_gmt":"2017-12-28T16:54:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=4902"},"modified":"2018-05-17T00:07:37","modified_gmt":"2018-05-17T00:07:37","slug":"linear-inequalities-in-one-variable","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/linear-inequalities-in-one-variable\/","title":{"raw":"Linear Inequalities in One Variable","rendered":"Linear Inequalities in One Variable"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use the addition and multiplication properties to solve algebraic inequalities<\/li>\r\n \t<li>Express solutions to inequalities graphically, with interval notation, and as an inequality<\/li>\r\n \t<li>Simplify and solve algebraic inequalities using the distributive property to clear\u00a0parentheses and fractions<\/li>\r\n<\/ul>\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 equalities. 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; doing so reverses the inequality symbol.\r\n\r\nThere are three ways to represent solutions to inequalities: an interval, a graph, and an inequality.\u00a0Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.\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\nThe following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Multiply By<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]ac&gt;bc[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]15&gt;9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex]-ac&lt;-bc[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-15&lt;-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following table illustrates how the division\u00a0property is applied to inequalities, and how dividing by a negative reverses the inequality:\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Divide By<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{a}{c}&gt;\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4&gt;2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{4}{2}&gt;\\frac{2}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex] \\displaystyle -\\frac{a}{c}&lt;-\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4&gt;2[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex] \\displaystyle -\\frac{4}{2}&lt;-\\frac{2}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"105622\"]\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<ol>\r\n \t<li>[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]<\/li>\r\n \t<li>[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]<\/li>\r\n \t<li>[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]<\/li>\r\n<\/ol>\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\"]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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"749552\"]\r\n<ol>\r\n \t<li>[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]<\/li>\r\n \t<li>[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]<\/li>\r\n \t<li>[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]<\/li>\r\n<\/ol>\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"32307\"][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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"453286\"]\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"703883\"][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\"]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\"]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>Simplify and solve algebraic inequalities using the distributive property<\/h2>\r\nAs with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>. [latex]2\\left(3x\u20135\\right)\\leq 4x+6[\/latex]\r\n\r\n[reveal-answer q=\"587737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587737\"]\r\n\r\nDistribute to clear the parentheses.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\,2(3x-5)\\leq 4x+6\\\\\\,\\,\\,\\,6x-10\\leq 4x+6\\end{array}[\/latex]<\/p>\r\nSubtract 4<i>x <\/i>from both sides to get the variable term on one side only.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}6x-10\\le 4x+6\\\\\\underline{-4x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-4x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,2x-10\\,\\,\\leq \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides to isolate the variable.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\\\\\,\\,\\,2x-10\\,\\,\\le \\,\\,\\,\\,\\,\\,\\,\\,6\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,+10\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\\\,\\,\\,2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,\\,\\,16\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides by 2 to express the variable with a coefficient of 1.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{2x}\\le \\,\\,\\,\\underline{16}\\\\\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\le \\,\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]x\\le8[\/latex]\r\nInterval: [latex]\\left(-\\infty,8\\right][\/latex]\r\nGraph: The graph of this solution set includes 8 and everything left of 8 on the number line.\r\n\r\n<img class=\"wp-image-3947 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10205137\/Screen-Shot-2016-05-10-at-1.51.18-PM-300x40.png\" alt=\"Number line with the interval (-oo,8] graphed\" width=\"443\" height=\"59\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"808701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808701\"]\r\n\r\nFirst, check the end point 8 in the related equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}2(3x-5)=4x+6\\,\\,\\,\\,\\,\\,\\\\2(3\\,\\cdot \\,8-5)=4\\,\\cdot \\,8+6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(24-5)=32+6\\,\\,\\,\\,\\,\\,\\\\2(19)=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\38=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, choose another solution and evaluate the inequality for that value to make sure it is a true statement.\u00a0Try 0.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}2(3\\,\\cdot \\,0-5)\\le 4\\,\\cdot \\,0+6?\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(-5)\\le 6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10\\le 6\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]x\\le8[\/latex] is the solution to\u00a0[latex]\\left(-\\infty,8\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.\r\nhttps:\/\/youtu.be\/vjZ3rQFVkh8\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]143594[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use the addition and multiplication properties to solve algebraic inequalities<\/li>\n<li>Express solutions to inequalities graphically, with interval notation, and as an inequality<\/li>\n<li>Simplify and solve algebraic inequalities using the distributive property to clear\u00a0parentheses and fractions<\/li>\n<\/ul>\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 equalities. 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; doing so reverses the inequality symbol.<\/p>\n<p>There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality.\u00a0Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.<\/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<p>The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Multiply By<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]ac>bc[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]15>9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]-ac<-bc[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-15<-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following table illustrates how the division\u00a0property is applied to inequalities, and how dividing by a negative reverses the inequality:<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Divide By<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{a}{c}>\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4>2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{4}{2}>\\frac{2}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]\\displaystyle -\\frac{a}{c}<-\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4>2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\displaystyle -\\frac{4}{2}<-\\frac{2}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.<\/p>\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\">Solution<\/span><\/p>\n<div id=\"q105622\" class=\"hidden-answer\" style=\"display: none\">\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.<\/p>\n<ol>\n<li>[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]<\/li>\n<li>[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]<\/li>\n<li>[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]<\/li>\n<\/ol>\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\">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\">Solution<\/span><\/p>\n<div id=\"q749552\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[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]<\/li>\n<li>[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]<\/li>\n<li>[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]<\/li>\n<\/ol>\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\">Solution<\/span><\/p>\n<div id=\"q32307\" class=\"hidden-answer\" style=\"display: none\">[latex]x\\ge -5[\/latex]<\/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\">Solution<\/span><\/p>\n<div id=\"q453286\" class=\"hidden-answer\" style=\"display: none\">\nSolving 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\">Solution<\/span><\/p>\n<div id=\"q703883\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(2,\\infty \\right)[\/latex]<\/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\">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\">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>Simplify and solve algebraic inequalities using the distributive property<\/h2>\n<p>As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>. [latex]2\\left(3x\u20135\\right)\\leq 4x+6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587737\">Show Solution<\/span><\/p>\n<div id=\"q587737\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute to clear the parentheses.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\,2(3x-5)\\leq 4x+6\\\\\\,\\,\\,\\,6x-10\\leq 4x+6\\end{array}[\/latex]<\/p>\n<p>Subtract 4<i>x <\/i>from both sides to get the variable term on one side only.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}6x-10\\le 4x+6\\\\\\underline{-4x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-4x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,2x-10\\,\\,\\leq \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6\\end{array}[\/latex]<\/p>\n<p>Add 10 to both sides to isolate the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\\\\\,\\,\\,2x-10\\,\\,\\le \\,\\,\\,\\,\\,\\,\\,\\,6\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,+10\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\\\,\\,\\,2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,\\,\\,16\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Divide both sides by 2 to express the variable with a coefficient of 1.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{2x}\\le \\,\\,\\,\\underline{16}\\\\\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\le \\,\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]x\\le8[\/latex]<br \/>\nInterval: [latex]\\left(-\\infty,8\\right][\/latex]<br \/>\nGraph: The graph of this solution set includes 8 and everything left of 8 on the number line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3947 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10205137\/Screen-Shot-2016-05-10-at-1.51.18-PM-300x40.png\" alt=\"Number line with the interval (-oo,8&#093; graphed\" width=\"443\" height=\"59\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q808701\">Show Solution<\/span><\/p>\n<div id=\"q808701\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, check the end point 8 in the related equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}2(3x-5)=4x+6\\,\\,\\,\\,\\,\\,\\\\2(3\\,\\cdot \\,8-5)=4\\,\\cdot \\,8+6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(24-5)=32+6\\,\\,\\,\\,\\,\\,\\\\2(19)=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\38=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement.\u00a0Try 0.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}2(3\\,\\cdot \\,0-5)\\le 4\\,\\cdot \\,0+6?\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(-5)\\le 6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10\\le 6\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]x\\le8[\/latex] is the solution to\u00a0[latex]\\left(-\\infty,8\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solve a Linear Inequality Requiring Multiple Steps (One Var)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vjZ3rQFVkh8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm143594\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143594&theme=oea&iframe_resize_id=ohm143594&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\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-4902\">\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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program.. <strong>Authored by<\/strong>:  . <strong>Provided by<\/strong>: Monterey Institute of Technology and Education.. <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>Authored by<\/strong>: Jay Abramson, et al... <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\">https:\/\/courses.candelalearning.com\/collegealgebra1xmaster<\/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>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><\/li><li>Solve a Linear Inequality Requiring Multiple Steps (One Var). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vjZ3rQFVkh8\">https:\/\/youtu.be\/vjZ3rQFVkh8<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":60342,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program.\",\"author\":\" \",\"organization\":\"Monterey Institute of Technology and Education.\",\"url\":\" http:\/\/nrocnetwork.org\/dm-opentext.\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson, et al..\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID# 92604, 92605, 92606, 92607, 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