2.5 - Solving Inequalities Algebraically and Graphically

2.5 - Solving Inequalities Algebraically and Graphically -...

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<?xml version="1.0" encoding="utf-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>2.5 - Solving Inequalities Algebraically and Graphically</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>2.5 - Solving Inequalities Algebraically and Graphically</h1> <h2>Linear Inequalities</h2> <p>When solving a linear inequality, treat it just like you were solving an equation with a few exceptions. </p> <ol> <li>When you multiply or divide both sides of an inequality by a negative constant, change the sense (direction) of the inequality.</li> <li>When both sides of an inequality are the same sign, change the sense of the inequality when you take the reciprocal of both sides.</li> <li>You may chain like inequalities together: <ul> <li>If a<b and b<c, then a<c (transitive property)</li> <li>If a&lt;b and c&lt;d, then a+c&lt;b+d. In English, that means that if you take two things that are smaller and put them together, it will be smaller than the two larger things put together. </li> </ul> </li> <li>You can not combine mixed inequalities. <ul> <li>If a&lt;b and b&gt;c, then you can't say for sure that a&lt;c or that c<a.</li> <li>If a&lt;b and c&gt;d, then you can't say for sure that a+c&lt;b+d or that a+c>b+d</li> </ul> </li> <li>If you rewrite the entire problem, just switching sides, make sure you change the sense of the inequality so that it still points to the same quantity.</li> <li>The following operations do <strong>not</strong> change the sense of the inequality <ul> <li>Adding a constant to both sides of an inequality</li> <li>Subtracting a constant from both sides of an inequality</li> <li>Multiplying or dividing both sides of an inequality by a positive constant</li> </ul> </li> </ol> <h2>Double Inequalities</h2> <p>Sometimes, two inequalities are combined into one. However, you need to be careful: </p> <p>If x&gt;3 and x&lt;6, then you can write 3&lt;x&lt;6. But, if x&lt;3 or
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x>6, you can <strong>not</strong> write 3>x>6 because that would imply that 3&gt;6 and that would be a false statement and the set would be empty. </p> <p>To solve a double inequality, just apply the operations to all three portions: If 3<x+2<6, then subtract 2 from all three parts to get 1<x<4.</p> <h2>Absolute Value Inequalities</h2> <p>These are going to give you trouble. They don't have to, but they will. People just don't get absolute values.</p> <blockquote>I could tell you that the major reason people don't get it is because they don't understand restrictions. That would be the truth, but just telling you that isn't going to help you. The book is going to give a short cut and people are going to ask &quot;why are there so many rules?&quot;. There aren't! There is one definition of absolute value, and if you know it, and apply the restrictions like you're supposed to, then the problems
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This note was uploaded on 10/18/2011 for the course MAT 1033 taught by Professor Brown during the Spring '10 term at Valencia.

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2.5 - Solving Inequalities Algebraically and Graphically -...

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