{"id":439,"date":"2015-10-26T18:34:06","date_gmt":"2015-10-26T18:34:06","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=439"},"modified":"2015-11-12T18:37:59","modified_gmt":"2015-11-12T18:37:59","slug":"understanding-compound-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/understanding-compound-inequalities\/","title":{"raw":"Understanding Compound Inequalities","rendered":"Understanding Compound Inequalities"},"content":{"raw":"A compound inequality<\/strong> includes two inequalities in one statement. A statement such as [latex]4<x\\le 6[\/latex] means [latex]4<x[\/latex] and [latex]x\\le 6[\/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.\r\n
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Example 7: Solving a Compound Inequality<\/h3>\r\nSolve the compound inequality: [latex]3\\le 2x+2<6[\/latex].\r\n\r\n<\/div>\r\n
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Solution<\/h3>\r\nThe first method is to write two separate inequalities: [latex]3\\le 2x+2[\/latex] and [latex]2x+2<6[\/latex]. We solve them independently.\r\n
[latex]\\begin{array}{lll}3\\le 2x+2\\hfill & \\text{and}\\hfill & 2x+2<6\\hfill \\\\ 1\\le 2x\\hfill & \\hfill & 2x<4\\hfill \\\\ \\frac{1}{2}\\le x\\hfill & \\hfill & x<2\\hfill \\end{array}[\/latex]<\/div>\r\nThen, we can rewrite the solution as a compound inequality, the same way the problem began.\r\n
[latex]\\frac{1}{2}\\le x<2[\/latex]<\/div>\r\nIn interval notation, the solution is written as [latex]\\left[\\frac{1}{2},2\\right)[\/latex].\r\n\r\nThe second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.\r\n
[latex]\\begin{array}{ll}3\\le 2x+2<6\\hfill & \\hfill \\\\ 1\\le 2x<4\\hfill & \\text{Isolate the variable term, and subtract 2 from all three parts}.\\hfill \\\\ \\frac{1}{2}\\le x<2\\hfill & \\text{Divide through all three parts by 2}.\\hfill \\end{array}[\/latex]<\/div>\r\nWe get the same solution: [latex]\\left[\\frac{1}{2},2\\right)[\/latex].\r\n\r\n<\/div>\r\n
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Try It 7<\/h3>\r\nSolve the compound inequality [latex]4<2x - 8\\le 10[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n
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Example 8: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\r\nSolve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[\/latex].\r\n\r\n<\/div>\r\n
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Solution<\/h3>\r\nLets try the first method. Write two inequalities:<\/strong>\r\n
[latex]\\begin{array}{lll}3+x> 7x - 2\\hfill & \\text{and}\\hfill & 7x - 2> 5x - 10\\hfill \\\\ 3> 6x - 2\\hfill & \\hfill & 2x - 2> -10\\hfill \\\\ 5> 6x\\hfill & \\hfill & 2x> -8\\hfill \\\\ \\frac{5}{6}> x\\hfill & \\hfill & x> -4\\hfill \\\\ x< \\frac{5}{6}\\hfill & \\hfill & -4< x\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is [latex]-4<x<\\frac{5}{6}[\/latex] or in interval notation [latex]\\left(-4,\\frac{5}{6}\\right)[\/latex]. Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right, as they appear on a number line.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 3<\/b>[\/caption]\r\n\r\n<\/div>\r\n
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Try It 8<\/h3>\r\nSolve the compound inequality: [latex]3y<4 - 5y<5+3y[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

A compound inequality<\/strong> includes two inequalities in one statement. A statement such as [latex]4\n

Example 7: Solving a Compound Inequality<\/h3>\n

Solve the compound inequality: [latex]3\\le 2x+2<6[\/latex].\n\n<\/div>\n

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Solution<\/h3>\n

The first method is to write two separate inequalities: [latex]3\\le 2x+2[\/latex] and [latex]2x+2<6[\/latex]. We solve them independently.\n\n\n

[latex]\\begin{array}{lll}3\\le 2x+2\\hfill & \\text{and}\\hfill & 2x+2<6\\hfill \\\\ 1\\le 2x\\hfill & \\hfill & 2x<4\\hfill \\\\ \\frac{1}{2}\\le x\\hfill & \\hfill & x<2\\hfill \\end{array}[\/latex]<\/div>\n

Then, we can rewrite the solution as a compound inequality, the same way the problem began.<\/p>\n

[latex]\\frac{1}{2}\\le x<2[\/latex]<\/div>\n

In interval notation, the solution is written as [latex]\\left[\\frac{1}{2},2\\right)[\/latex].<\/p>\n

The second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the same time.<\/p>\n

[latex]\\begin{array}{ll}3\\le 2x+2<6\\hfill & \\hfill \\\\ 1\\le 2x<4\\hfill & \\text{Isolate the variable term, and subtract 2 from all three parts}.\\hfill \\\\ \\frac{1}{2}\\le x<2\\hfill & \\text{Divide through all three parts by 2}.\\hfill \\end{array}[\/latex]<\/div>\n

We get the same solution: [latex]\\left[\\frac{1}{2},2\\right)[\/latex].<\/p>\n<\/div>\n

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Try It 7<\/h3>\n

Solve the compound inequality [latex]4<2x - 8\\le 10[\/latex].\n\nSolution<\/a><\/p>\n<\/div>\n

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Example 8: Solving a Compound Inequality with the Variable in All Three Parts<\/h3>\n

Solve the compound inequality with variables in all three parts: [latex]3+x>7x - 2>5x - 10[\/latex].<\/p>\n<\/div>\n

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Solution<\/h3>\n

Lets try the first method. Write two inequalities:<\/strong><\/p>\n

[latex]\\begin{array}{lll}3+x> 7x - 2\\hfill & \\text{and}\\hfill & 7x - 2> 5x - 10\\hfill \\\\ 3> 6x - 2\\hfill & \\hfill & 2x - 2> -10\\hfill \\\\ 5> 6x\\hfill & \\hfill & 2x> -8\\hfill \\\\ \\frac{5}{6}> x\\hfill & \\hfill & x> -4\\hfill \\\\ x< \\frac{5}{6}\\hfill & \\hfill & -4< x\\hfill \\end{array}[\/latex]<\/div>\n

The solution set is [latex]-4\"A<\/p>\n

Figure 3<\/b><\/p>\n<\/div>\n<\/div>\n

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Try It 8<\/h3>\n

Solve the compound inequality: [latex]3y<4 - 5y<5+3y[\/latex].\n\nSolution<\/a><\/p>\n<\/div>\n\n\t\t\t

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