{"id":16146,"date":"2019-10-01T18:18:40","date_gmt":"2019-10-01T18:18:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-solve-single-step-inequalities-2\/"},"modified":"2020-10-22T09:12:54","modified_gmt":"2020-10-22T09:12:54","slug":"read-solve-single-step-inequalities-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-solve-single-step-inequalities-2\/","title":{"raw":"8.1.b - Solving Inequalities","rendered":"8.1.b &#8211; Solving Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"text-align: center\">Learning Outcomes<\/h3>\r\n<ul style=\"text-align: left\">\r\n \t<li>Solve single-step inequalities<\/li>\r\n \t<li>Solve multi-step inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 style=\"text-align: left\">Multiplication and Division Properties of Inequality<\/h2>\r\nSolving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. 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 similar 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; doing so reverses the inequality symbol.\r\n\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<\/h3>\r\nIllustrate the multiplication property for inequalities by solving each of the following:\r\n<ol style=\"list-style-type: lower-alpha\">\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=\"432848\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"432848\"]\r\n\r\na.\r\n[latex]\\begin{array}{cc}\\hfill3x&lt;6 \\hfill\\\\\\dfrac{1}{3}\\normalsize\\left(3x\\right)&lt;\\left(6\\right)\\dfrac{1}{3} \\\\ \\hfill{x}&lt;2 \\hfill\\end{array}[\/latex]\r\n\r\n&nbsp;\r\n\r\nb.\r\n[latex]\\begin{array}{rr}-2x - 1\\ge 5\\\\ \\hfill\\hfill-2x\\ge 6\\end{array}[\/latex]\r\n\r\nMultiply both sides by [latex]-\\dfrac{1}{2}[\/latex].\r\n\r\n[latex]\\begin{array}{ll}\\hfill\\hfill\\left(-\\dfrac{1}{2}\\normalsize\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\dfrac{1}{2}\\normalsize\\right)\\end{array}[\/latex]\r\n\r\nReverse the inequality.\r\n\r\n[latex]\\begin{array}{l}\\hfill&amp;\\hfill&amp;\\hfill&amp;\\hfill&amp;\\hfill x\\le -3\\end{array}[\/latex]\r\n\r\nc.\r\n[latex]\\begin{array}{ll}5-x&gt;10\\\\ -x&gt;5\\hfill &amp;\\hfill\\end{array}[\/latex]\r\n\r\nMultiply both sides by [latex] -1[\/latex].\r\n\r\n[latex]\\left(-1\\right)\\left(-x\\right)&gt;\\left(5\\right)\\left(-1\\right)[\/latex]\r\n\r\nReverse the inequality\r\n\r\n[latex]x&lt;-5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 style=\"text-align: left\">Solve Inequalities Using the Addition Property<\/h2>\r\nWhen we solve equations, we may need to add or subtract in order to isolate the variable; the same is true for inequalities. There are no special behaviors to watch out for when using the addition property to solve inequalities.\r\n\r\nThe following table illustrates how the addition property applies to inequalities.\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Add<\/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]a+c&gt;b+c[\/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]8&gt;6[\/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]a-c&gt;b-c[\/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]2&gt;0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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\nIn our next example, we will use the addition property to solve inequalities.\r\n<div class=\"textbox exercises\" style=\"text-align: left\">\r\n<h3>Example<\/h3>\r\nIllustrate the addition property for inequalities by solving each of the following:\r\n<ol style=\"list-style-type: lower-alpha\">\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=\"399605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399605\"]\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\na.\r\n[latex]\\begin{array}{rr}\\hfill x - 15&lt;4\\hfill\\hfill \\\\ \\hfill x - 15+15&lt;4+15\\hfill&amp; \\text{Add 15 to both sides.}\\hfill\\\\\\hfill\\quad x&lt;19 \\hfill\\end{array}[\/latex]\r\n\r\nb.\r\n[latex]\\begin{array}{rr}\\hfill 6\u2265 x - 1\\hfill\\hfill \\\\\\hfill 6+1\\ge x - 1+1\\hfill &amp; \\text{Add 1 to both sides}.\\hfill \\\\\\quad\\quad 7\u2265 x\\hfill \\end{array}[\/latex]\r\n\r\nc.\r\n[latex]\\begin{array}{rr}\\hfill x+7&gt;9\\hfill\\hfill\\\\\\hfill x+7 - 7&gt;9 - 7\\hfill &amp; \\text{Subtract 7 from both sides}.\\hfill\\quad \\\\\\hfill x&gt;2\\hfill \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of solving single-step inequalities using the multiplication and addition properties.\r\n\r\n[embed]https:\/\/youtu.be\/1Z22Xh66VFM[\/embed]\r\n\r\nThe following video shows examples of solving inequalities with the variable on the right side.\r\n\r\n[embed]https:\/\/youtu.be\/RBonYKvTCLU[\/embed]\r\n\r\n[ohm_question]77757[\/ohm_question]\r\n<h2 style=\"text-align: left\">Solve Multi-Step Inequalities<\/h2>\r\nAs the previous examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations. To isolate the variable and solve,\u00a0we combine like terms and perform operations with the multiplication and addition properties.\r\n<div class=\"textbox exercises\" style=\"text-align: left\">\r\n<h3>Example<\/h3>\r\nSolve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].\r\n\r\n[reveal-answer q=\"532189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"532189\"]\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}{rr}13 - 7x\\ge 10x - 4\\hfill &amp; \\\\ 13 - 17x\\ge -4\\hfill &amp; \\text{Move variable terms to one side of the inequality}.\\hfill&amp;\\quad \\\\-17x\\ge -17\\hfill&amp;\\text{Isolate the variable term}.\\hfill&amp;\\quad \\\\x\\le 1\\hfill &amp; \\text{Dividing both sides by -17 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\n[ohm_question]143594[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next example, we solve an inequality that contains fractions. Notice how we need to reverse the inequality sign at the end because we multiply by a negative.\r\n<div class=\"textbox exercises\" style=\"text-align: left\">\r\n<h3>Example<\/h3>\r\nSolve the following inequality and write the answer in interval notation: [latex]-\\dfrac{3}{4}\\normalsize x\\ge -\\dfrac{5}{8}\\normalsize +\\dfrac{2}{3}\\normalsize x[\/latex].\r\n\r\n[reveal-answer q=\"59887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"59887\"]\r\n\r\nWe begin solving in the same way we do when solving an equation.\r\n<div style=\"text-align: center\">[latex]\\begin{array}{rr}-\\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\n<div>The solution set is the interval [latex]\\left(-\\infty ,\\dfrac{15}{34}\\normalsize\\right][\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"text-align: center\">Learning Outcomes<\/h3>\n<ul style=\"text-align: left\">\n<li>Solve single-step inequalities<\/li>\n<li>Solve multi-step inequalities<\/li>\n<\/ul>\n<\/div>\n<h2 style=\"text-align: left\">Multiplication and Division Properties of Inequality<\/h2>\n<p>Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. 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 similar 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; doing so reverses the inequality symbol.<\/p>\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<\/h3>\n<p>Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\n<ol style=\"list-style-type: lower-alpha\">\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=\"q432848\">Show Solution<\/span><\/p>\n<div id=\"q432848\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.<br \/>\n[latex]\\begin{array}{cc}\\hfill3x<6 \\hfill\\\\\\dfrac{1}{3}\\normalsize\\left(3x\\right)<\\left(6\\right)\\dfrac{1}{3} \\\\ \\hfill{x}<2 \\hfill\\end{array}[\/latex]\n\n&nbsp;\n\nb.\n[latex]\\begin{array}{rr}-2x - 1\\ge 5\\\\ \\hfill\\hfill-2x\\ge 6\\end{array}[\/latex]\n\nMultiply both sides by [latex]-\\dfrac{1}{2}[\/latex].\n\n[latex]\\begin{array}{ll}\\hfill\\hfill\\left(-\\dfrac{1}{2}\\normalsize\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\dfrac{1}{2}\\normalsize\\right)\\end{array}[\/latex]\n\nReverse the inequality.\n\n[latex]\\begin{array}{l}\\hfill&\\hfill&\\hfill&\\hfill&\\hfill x\\le -3\\end{array}[\/latex]\n\nc.\n[latex]\\begin{array}{ll}5-x>10\\\\ -x>5\\hfill &\\hfill\\end{array}[\/latex]<\/p>\n<p>Multiply both sides by [latex]-1[\/latex].<\/p>\n<p>[latex]\\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)[\/latex]<\/p>\n<p>Reverse the inequality<\/p>\n<p>[latex]x<-5[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: left\">Solve Inequalities Using the Addition Property<\/h2>\n<p>When we solve equations, we may need to add or subtract in order to isolate the variable; the same is true for inequalities. There are no special behaviors to watch out for when using the addition property to solve inequalities.<\/p>\n<p>The following table illustrates how the addition property applies to inequalities.<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Add<\/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]a+c>b+c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]8>6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]a-c>b-c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]2>0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These properties also apply to [latex]a\\le b[\/latex], [latex]a>b[\/latex], and [latex]a\\ge b[\/latex].<\/p>\n<p>In our next example, we will use the addition property to solve inequalities.<\/p>\n<div class=\"textbox exercises\" style=\"text-align: left\">\n<h3>Example<\/h3>\n<p>Illustrate the addition property for inequalities by solving each of the following:<\/p>\n<ol style=\"list-style-type: lower-alpha\">\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=\"q399605\">Show Solution<\/span><\/p>\n<div id=\"q399605\" 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.<br \/>\na.<br \/>\n[latex]\\begin{array}{rr}\\hfill x - 15<4\\hfill\\hfill \\\\ \\hfill x - 15+15<4+15\\hfill& \\text{Add 15 to both sides.}\\hfill\\\\\\hfill\\quad x<19 \\hfill\\end{array}[\/latex]\n\nb.\n[latex]\\begin{array}{rr}\\hfill 6\u2265 x - 1\\hfill\\hfill \\\\\\hfill 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\\\quad\\quad 7\u2265 x\\hfill \\end{array}[\/latex]\n\nc.\n[latex]\\begin{array}{rr}\\hfill x+7>9\\hfill\\hfill\\\\\\hfill x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill\\quad \\\\\\hfill x>2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of solving single-step inequalities using the multiplication and addition properties.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solving One Step Inequalities by Adding and Subtracting (Variable Left Side)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1Z22Xh66VFM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following video shows examples of solving inequalities with the variable on the right side.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Solving One Step Inequalities by Adding and Subtracting (Variable Right Side)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RBonYKvTCLU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm77757\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=77757&theme=oea&iframe_resize_id=ohm77757&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<h2 style=\"text-align: left\">Solve Multi-Step Inequalities<\/h2>\n<p>As the previous examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations. To isolate the variable and solve,\u00a0we combine like terms and perform operations with the multiplication and addition properties.<\/p>\n<div class=\"textbox exercises\" style=\"text-align: left\">\n<h3>Example<\/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=\"q532189\">Show Solution<\/span><\/p>\n<div id=\"q532189\" 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}{rr}13 - 7x\\ge 10x - 4\\hfill & \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill&\\quad \\\\-17x\\ge -17\\hfill&\\text{Isolate the variable term}.\\hfill&\\quad \\\\x\\le 1\\hfill & \\text{Dividing both sides by -17 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><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>In the next example, we solve an inequality that contains fractions. Notice how we need to reverse the inequality sign at the end because we multiply by a negative.<\/p>\n<div class=\"textbox exercises\" style=\"text-align: left\">\n<h3>Example<\/h3>\n<p>Solve the following inequality and write the answer in interval notation: [latex]-\\dfrac{3}{4}\\normalsize x\\ge -\\dfrac{5}{8}\\normalsize +\\dfrac{2}{3}\\normalsize x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q59887\">Show Solution<\/span><\/p>\n<div id=\"q59887\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin solving in the same way we do when solving an equation.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}{rr}-\\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<div>The solution set is the interval [latex]\\left(-\\infty ,\\dfrac{15}{34}\\normalsize\\right][\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\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-16146\">\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>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>Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/1Z22Xh66VFM\">https:\/\/youtu.be\/1Z22Xh66VFM<\/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>Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RBonYKvTCLU\">https:\/\/youtu.be\/RBonYKvTCLU<\/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>Ex: Solve One Step Linear Inequality by Dividing (Variable Left). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IajiD3R7U-0\">https:\/\/youtu.be\/IajiD3R7U-0<\/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>Ex: Solve One Step Linear Inequality by Dividing (Variable Right). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/s9fJOnVTHhs\">https:\/\/youtu.be\/s9fJOnVTHhs<\/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":169554,"menu_order":4,"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 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