{"id":4553,"date":"2017-06-07T18:51:06","date_gmt":"2017-06-07T18:51:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-solve-multi-step-inequalities\/"},"modified":"2017-08-15T15:27:11","modified_gmt":"2017-08-15T15:27:11","slug":"read-solve-multi-step-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-solve-multi-step-inequalities\/","title":{"raw":"Solve Multi-Step Inequalities","rendered":"Solve Multi-Step Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Solve multi-step inequalities\r\n<ul>\r\n \t<li>Combine properties of inequality to isolate variables,\u00a0solve algebraic\u00a0inequalities, and express their solutions graphically<\/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<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Combine properties of inequality to \u00a0solve algebraic\u00a0inequalities<\/h2>\r\nA popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and\/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>p<\/i>. [latex]4p+5&lt;29[\/latex]\r\n\r\n[reveal-answer q=\"211828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211828\"]\r\n\r\nBegin to isolate the variable by subtracting 5 from both sides of the inequality.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}4p+5&lt;\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\4p\\,\\,\\,\\,\\,\\,\\,\\,\\,&lt;\\,\\,24\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by 4 to express the variable with a coefficient of 1.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\underline{4p}\\,&lt;\\,\\,\\underline{24}\\,\\,\\\\\\,4\\,\\,\\,\\,&lt;\\,\\,4\\\\\\,\\,\\,\\,\\,p&lt;6\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality:\u00a0[latex]p&lt;6[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,6\\right)[\/latex]\r\n\r\nGraph: Note the\u00a0open circle at the end point 6 to show that solutions to the inequality do not include 6.\u00a0The values where <i>p<\/i> is less than 6 are found all along the number line to the left of 6.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185052\/image057.jpg\" alt=\"Number line. Open circle on 6. Highlight on every number less than 6.\" width=\"575\" height=\"31\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"291597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"291597\"]\r\n\r\nCheck the end point 6 in the related equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}4p+5=29\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4(6)+5=29?\\\\24+5=29\\,\\,\\,\\\\29=29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nTry another value to check the inequality. Let\u2019s use [latex]p=0[\/latex].\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}4p+5&lt;29\\,\\,\\,\\\\\\text{Is}\\,\\,\\,4(0)+5&lt;29?\\\\0+5&lt;29\\,\\,\\,\\\\5&lt;29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]p&lt;6[\/latex] is the solution to\u00a0[latex]4p+5&lt;29[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>: \u00a0[latex]3x\u20137\\ge 41[\/latex]\r\n[reveal-answer q=\"238157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238157\"]\r\n\r\nBegin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}3x-7\\ge 41\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+7\\,\\,\\,\\,+7}\\\\\\frac{3x}{3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{48}{3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 16\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]x\\ge 16[\/latex]\r\n\r\nInterval: [latex]\\left[16,\\infty\\right)[\/latex]\r\n\r\nGraph:\u00a0To graph this inequality, you draw a closed circle at the end point 16 on the number line\u00a0to show that solutions include the value 16. The line continues to the right from 16 because all the numbers greater than 16 will also make the inequality\u00a0[latex]3x\u20137\\ge 41[\/latex] true.\r\n<img class=\"aligncenter wp-image-3956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185054\/Screen-Shot-2016-05-10-at-4.28.03-PM-300x48.png\" alt=\"Closed dot on 16, line through all numbers greater than 16.\" width=\"425\" height=\"68\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"437341\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"437341\"]\r\n\r\nFirst, check the end point 16 in the related equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}3x-7=41\\,\\,\\,\\\\\\text{Does}\\,\\,\\,3(16)-7=41?\\\\48-7=41\\,\\,\\,\\\\41=41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Let\u2019s use [latex]x = 20[\/latex].\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,3x-7\\ge 41\\,\\,\\,\\\\\\text{Is}\\,\\,\\,\\,\\,3(20)-7\\ge 41?\\\\60-7\\ge 41\\,\\,\\,\\\\53\\ge 41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>p<\/i>. [latex]\u221258&gt;14\u22126p[\/latex]\r\n\r\n[reveal-answer q=\"424351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"424351\"]\r\n\r\nNote how the variable is on the right hand side of the inequality, the method for solving does not change in this case.\r\n\r\nBegin to isolate the variable by subtracting 14 from both sides of the inequality.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{l}\u221258\\,\\,&gt;14\u22126p\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-14\\,\\,\\,\\,\\,\\,\\,-14}\\\\-72\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&gt;-6p\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by [latex]\u22126[\/latex] to express the variable with a coefficient of 1.\u00a0Dividing by a negative number results in reversing the inequality sign.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\underline{-72}&gt;\\underline{-6p}\\\\-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\\\\\,\\,\\,\\,\\,\\,12\\lt{p}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">We can also write this as [latex]p&gt;12[\/latex]. \u00a0 Notice how the inequality sign is still opening up toward the variable p.<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]p&gt;12[\/latex]\r\nInterval: [latex]\\left(12,\\infty\\right)[\/latex]\r\nGraph: The graph of the inequality <i>p <\/i>&gt; 12 has an open circle at 12 with an arrow stretching to the right.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185056\/image059.jpg\" alt=\"Number line. Open circle on 12. Highlight on all numbers over 12.\" width=\"575\" height=\"31\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"500309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500309\"]\r\n\r\nFirst, check the end point 12 in the related equation.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}-58=14-6p\\\\-58=14-6\\left(12\\right)\\\\-58=14-72\\\\-58=-58\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Try 100.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}-58&gt;14-6p\\\\-58&gt;14-6\\left(100\\right)\\\\-58&gt;14-600\\\\-58&gt;-586\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see an example of solving a linear inequality with the variable on the\u00a0left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.\r\n\r\nhttps:\/\/youtu.be\/RB9wvIogoEM\r\n\r\nIn the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.\r\n\r\nhttps:\/\/youtu.be\/9D2g_FaNBkY\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\/2011\/2017\/06\/07185059\/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 key-takeaways\">\r\n<h3>Think\u00a0About It<\/h3>\r\nIn the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.\r\n\r\nSolve for a. [latex] \\displaystyle\\frac{{2}{a}-{4}}{{6}}{&lt;2}[\/latex]\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"701072\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"701072\"]\r\n\r\nClear the fraction by multiplying both sides of the equation by 6.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\frac{{2}{a}-{4}}{{6}}{&lt;2}\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\6\\,\\cdot \\,\\frac{2a-4}{6}&lt;2\\,\\cdot \\,6\\\\\\\\{2a-4}&lt;12\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 4 to both sides to isolate the variable.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}2a-4&lt;12\\\\\\underline{\\,\\,\\,+4\\,\\,\\,\\,+4}\\\\2a&lt;16\\end{array}[\/latex]<\/p>\r\n<span style=\"line-height: 1.5\">Divide both sides by 2 to express the variable with a coefficient of 1.<\/span>\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{c}\\frac{2a}{2}&lt;\\,\\frac{16}{2}\\\\\\\\a&lt;8\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]a&lt;8[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,8\\right)[\/latex]\r\n\r\nGraph: The graph of this solution contains a solid dot at 8 to show that 8 is included in the solution set. The line continues to the left to show that values less than 8 are also included in the solution set.<img class=\"aligncenter wp-image-3948\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185103\/Screen-Shot-2016-05-10-at-1.54.23-PM-300x32.png\" alt=\"Open circle on 8 and line through all numbers less than 8.\" width=\"469\" height=\"50\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"905072\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"905072\"]\r\nFirst, check the end point 8 in the related equation.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\frac{2a-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\text{Does}\\,\\,\\,\\frac{2(8)-4}{6}=2?\\\\\\\\\\frac{16-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\frac{12}{6}=2\\,\\,\\,\\,\\\\\\\\2=2\\,\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 5.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\text{Is}\\,\\,\\,\\frac{2(5)-4}{6}&lt;2?\\\\\\\\\\frac{10-4}{6}&lt;2\\,\\,\\,\\\\\\\\\\,\\,\\,\\,\\frac{6}{6}&lt;2\\,\\,\\,\\\\\\\\1&lt;2\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nInequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract <i>either positive or negative<\/i> numbers to both sides of the inequality.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Solve multi-step inequalities\n<ul>\n<li>Combine properties of inequality to isolate variables,\u00a0solve algebraic\u00a0inequalities, and express their solutions graphically<\/li>\n<li>Simplify and solve algebraic inequalities using the distributive property to clear\u00a0parentheses and fractions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Combine properties of inequality to \u00a0solve algebraic\u00a0inequalities<\/h2>\n<p>A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and\/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>p<\/i>. [latex]4p+5<29[\/latex]\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211828\">Show Solution<\/span><\/p>\n<div id=\"q211828\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin to isolate the variable by subtracting 5 from both sides of the inequality.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\4p\\,\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,24\\,\\,\\end{array}[\/latex]<\/p>\n<p>Divide both sides of the inequality by 4 to express the variable with a coefficient of 1.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\underline{4p}\\,<\\,\\,\\underline{24}\\,\\,\\\\\\,4\\,\\,\\,\\,<\\,\\,4\\\\\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality:\u00a0[latex]p<6[\/latex]\n\nInterval: [latex]\\left(-\\infty,6\\right)[\/latex]\n\nGraph: Note the\u00a0open circle at the end point 6 to show that solutions to the inequality do not include 6.\u00a0The values where <i>p<\/i> is less than 6 are found all along the number line to the left of 6.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185052\/image057.jpg\" alt=\"Number line. Open circle on 6. Highlight on every number less than 6.\" width=\"575\" height=\"31\" \/><\/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=\"q291597\">Show Solution<\/span><\/p>\n<div id=\"q291597\" class=\"hidden-answer\" style=\"display: none\">\n<p>Check the end point 6 in the related equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}4p+5=29\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4(6)+5=29?\\\\24+5=29\\,\\,\\,\\\\29=29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Try another value to check the inequality. Let\u2019s use [latex]p=0[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}4p+5<29\\,\\,\\,\\\\\\text{Is}\\,\\,\\,4(0)+5<29?\\\\0+5<29\\,\\,\\,\\\\5<29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]p<6[\/latex] is the solution to\u00a0[latex]4p+5<29[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>: \u00a0[latex]3x\u20137\\ge 41[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q238157\">Show Solution<\/span><\/p>\n<div id=\"q238157\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}3x-7\\ge 41\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+7\\,\\,\\,\\,+7}\\\\\\frac{3x}{3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{48}{3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 16\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]x\\ge 16[\/latex]<\/p>\n<p>Interval: [latex]\\left[16,\\infty\\right)[\/latex]<\/p>\n<p>Graph:\u00a0To graph this inequality, you draw a closed circle at the end point 16 on the number line\u00a0to show that solutions include the value 16. The line continues to the right from 16 because all the numbers greater than 16 will also make the inequality\u00a0[latex]3x\u20137\\ge 41[\/latex] true.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185054\/Screen-Shot-2016-05-10-at-4.28.03-PM-300x48.png\" alt=\"Closed dot on 16, line through all numbers greater than 16.\" width=\"425\" height=\"68\" \/><\/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=\"q437341\">Show Solution<\/span><\/p>\n<div id=\"q437341\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, check the end point 16 in the related equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}3x-7=41\\,\\,\\,\\\\\\text{Does}\\,\\,\\,3(16)-7=41?\\\\48-7=41\\,\\,\\,\\\\41=41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Then, try another value to check the inequality. Let\u2019s use [latex]x = 20[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\,3x-7\\ge 41\\,\\,\\,\\\\\\text{Is}\\,\\,\\,\\,\\,3(20)-7\\ge 41?\\\\60-7\\ge 41\\,\\,\\,\\\\53\\ge 41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>p<\/i>. [latex]\u221258>14\u22126p[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q424351\">Show Solution<\/span><\/p>\n<div id=\"q424351\" class=\"hidden-answer\" style=\"display: none\">\n<p>Note how the variable is on the right hand side of the inequality, the method for solving does not change in this case.<\/p>\n<p>Begin to isolate the variable by subtracting 14 from both sides of the inequality.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}\u221258\\,\\,>14\u22126p\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-14\\,\\,\\,\\,\\,\\,\\,-14}\\\\-72\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>-6p\\end{array}[\/latex]<\/p>\n<p>Divide both sides of the inequality by [latex]\u22126[\/latex] to express the variable with a coefficient of 1.\u00a0Dividing by a negative number results in reversing the inequality sign.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\underline{-72}>\\underline{-6p}\\\\-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\\\\\,\\,\\,\\,\\,\\,12\\lt{p}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">We can also write this as [latex]p>12[\/latex]. \u00a0 Notice how the inequality sign is still opening up toward the variable p.<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]p>12[\/latex]<br \/>\nInterval: [latex]\\left(12,\\infty\\right)[\/latex]<br \/>\nGraph: The graph of the inequality <i>p <\/i>&gt; 12 has an open circle at 12 with an arrow stretching to the right.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185056\/image059.jpg\" alt=\"Number line. Open circle on 12. Highlight on all numbers over 12.\" width=\"575\" height=\"31\" \/><\/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=\"q500309\">Show Solution<\/span><\/p>\n<div id=\"q500309\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, check the end point 12 in the related equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}-58=14-6p\\\\-58=14-6\\left(12\\right)\\\\-58=14-72\\\\-58=-58\\end{array}[\/latex]<\/p>\n<p>Then, try another value to check the inequality. Try 100.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}-58>14-6p\\\\-58>14-6\\left(100\\right)\\\\-58>14-600\\\\-58>-586\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see an example of solving a linear inequality with the variable on the\u00a0left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solve a Two Step Linear Inequality  (Variable Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RB9wvIogoEM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Solve a Two Step Linear Inequality  (Variable Right)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9D2g_FaNBkY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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\/2011\/2017\/06\/07185059\/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-3\" 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 key-takeaways\">\n<h3>Think\u00a0About It<\/h3>\n<p>In the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.<\/p>\n<p>Solve for a. [latex]\\displaystyle\\frac{{2}{a}-{4}}{{6}}{<2}[\/latex]\n\n<textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q701072\">Show Solution<\/span><\/p>\n<div id=\"q701072\" class=\"hidden-answer\" style=\"display: none\">\n<p>Clear the fraction by multiplying both sides of the equation by 6.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\frac{{2}{a}-{4}}{{6}}{<2}\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\6\\,\\cdot \\,\\frac{2a-4}{6}<2\\,\\cdot \\,6\\\\\\\\{2a-4}<12\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add 4 to both sides to isolate the variable.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}2a-4<12\\\\\\underline{\\,\\,\\,+4\\,\\,\\,\\,+4}\\\\2a<16\\end{array}[\/latex]<\/p>\n<p><span style=\"line-height: 1.5\">Divide both sides by 2 to express the variable with a coefficient of 1.<\/span><\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{c}\\frac{2a}{2}<\\,\\frac{16}{2}\\\\\\\\a<8\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]a<8[\/latex]\n\nInterval: [latex]\\left(-\\infty,8\\right)[\/latex]\n\nGraph: The graph of this solution contains a solid dot at 8 to show that 8 is included in the solution set. The line continues to the left to show that values less than 8 are also included in the solution set.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3948\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185103\/Screen-Shot-2016-05-10-at-1.54.23-PM-300x32.png\" alt=\"Open circle on 8 and line through all numbers less than 8.\" width=\"469\" height=\"50\" \/><\/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=\"q905072\">Show Solution<\/span><\/p>\n<div id=\"q905072\" class=\"hidden-answer\" style=\"display: none\">\nFirst, check the end point 8 in the related equation.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\frac{2a-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\text{Does}\\,\\,\\,\\frac{2(8)-4}{6}=2?\\\\\\\\\\frac{16-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\frac{12}{6}=2\\,\\,\\,\\,\\\\\\\\2=2\\,\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\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. Try 5.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\text{Is}\\,\\,\\,\\frac{2(5)-4}{6}<2?\\\\\\\\\\frac{10-4}{6}<2\\,\\,\\,\\\\\\\\\\,\\,\\,\\,\\frac{6}{6}<2\\,\\,\\,\\\\\\\\1<2\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>Inequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract <i>either positive or negative<\/i> numbers to both sides of the inequality.<\/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-4553\">\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>Ex: Solve a Two Step Linear Inequality (Variable Left). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RB9wvIogoEM\">https:\/\/youtu.be\/RB9wvIogoEM<\/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>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: Solve a Two Step Linear Inequality (Variable Right). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9D2g_FaNBkY\">https:\/\/youtu.be\/9D2g_FaNBkY<\/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 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":17533,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Solve a Two Step Linear Inequality (Variable Left)\",\"author\":\"James Sousa (Mathispower4u.com) for 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