o Section 4: Compound Inequalities

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 1.4 Compound Inequalities 53 Version: Fall 2007 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities , which is a phrase that usually refers to a pair of inequalities connected either by the word “and” or the word “or.” Before we begin with the advanced work of solving these inequalities, let’s first spend a word or two (for purposes of review) discussing the solution of simple linear inequalities. Simple Linear Inequalities As in solving equations, you may add or subtract the same amount from both sides of an inequality. Property 1. Let a and b be real numbers with a < b . If c is any real number, then a + c < b + c and a c < b c. This utility is equally valid if you replace the “less than” symbol with > , , or . Example 2. Solve the inequality x + 3 < 8 for x . Subtract 3 from both sides of the inequality and simplify. x + 3 < 8 x + 3 3 < 8 3 x < 5 Thus, all real numbers less than 5 are solutions of the inequality. It is traditional to sketch the solution set of inequalities on a number line. 5 We can describe the solution set using set-builder and interval notation. The solu- tion is ( −∞ , 5) = { x : x < 5 } . An important concept is the idea of equivalent inequalities . Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1
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54 Chapter 1 Preliminaries Version: Fall 2007 Equivalent Inequalities. Two inequalities are said to be equivalent if and only if they have the same solution set. Note that this definition is similar to the definition of equivalent equations. That is, two inequalities are equivalent if all of the solutions of the first inequality are also solutions of the second inequality, and vice-versa. Thus, in Example 2 , subtracting three from both sides of the original inequality produced an equivalent inequality. That is, the inequalities x +3 < 8 and x < 5 have the same solution set, namely, all real numbers that are less than 5. It is no coincidence that the tools in Property 1 produce equivalent inequalities. Whenver you add or subtract the same amount from both sides of an inequality, the resulting inequality is equivalent to the original (they have the same solution set). Let’s look at another example. Example 3. Solve the inequality x 5 4 for x . Add 5 to both sides of the inequality and simplify. x 5 4 x 5 + 5 4 + 5 x 9 Shade the solution on a number line. 9 In set-builder and interval notation, the solution is [9 , ) = { x : x 9 } You can also multiply or divide both sides by the same positive number. Property 4. Let a and b be real numbers with a < b . If c is a real positive number, then ac < bc and a c < b c . Again, this utility is equally valid if you replace the “less than” symbol by
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o Section 4: Compound Inequalities - Section 1.4 Compound...

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