Section 2.8 Linear Inequalities 191 Solution: To solve the compound inequality isolate the variable x in the middle. The operations performed on the middle portion of the inequality must also be performed on the left-hand side and right-hand side. Subtract 1 from all three parts of the inequality. Simplify. Divide by 2 in all three parts of the inequality. 3 2x 1 6 7 3 1 2x 1 1 6 7 1 4 2x 6 6 6 5 4 3 2 1 0 1 2 3 4 5 6 5x ƒ 2 x 636 4 2 2x 2 6 Set-builder notation: Interval notation: 32, 32 ( 2 x 6 3 6 2 3 2x 1 6 7 6. Applications of Linear Inequalities Table 2-2 provides several commonly used translations to express inequalities. Table 2-2 English Phrase Mathematical Inequality a is less than b a 6 b a is greater than b a b a exceeds b 7 a is less than or equal to b a is at most b a b a is no more than b a is greater than or equal to b a is at least b a b a is no less than b Skill Practice Translate the English phrase into a mathematical inequality. 15. Bill needs a score of at least 92 on the final exam. Let x represent Bill’s score. 16. Fewer than 19 cars are in the parking lot. Let c represent the number of cars. 17. The heights, h, of women who wear petite size clothing are typically between 58 in. and 63 in., inclusive. Answers 15. 16. 17. 58 h 63 x 92 c 6 19 Translating Expressions Involving Inequalities Example 9 Translate the English phrases into mathematical inequalities. a. Claude’s annual salary, s, is no more than $40,000. b. A citizen must be at least 18 years old to vote. (Let a represent a citizen’s age.) c. An amusement park ride has a height requirement between 48 in. and 70 in. (Let h represent height in inches.) Solution: a. Claude’s annual salary, s, is no more than $40,000. b. A citizen must be at least 18 years old to vote. c. An amusement park ride has a height requirement between 48 in. and 70 in. s 40,000 a 18 48 6 h 6 70 Linear inequalities are found in a variety of applications. Example 10 can help you determine the minimum grade you need on an exam to get an A in your math course. Classroom Example: p. 196, Exercises 106 and 108
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