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Properties of Inequalities: Multiplication Property of Inequality: Multiplying each side of an inequality by a positive number produces an equivalent inequality. Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. If a 0, then ac < bc. If a bc. Lesson 3.5

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Properties of Inequalities: Division Property of Inequality: Dividing each side of an inequality by a positive number produces an equivalent inequality. Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality. If a 0, then a/c < b/c If a b/c. Lesson 3.5

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EXAMPLE 1 Solving an Inequality Using Multiplication Original inequality Multiply each side by –8. Reverse inequality symbol. Simplify. n ≤ –16 1 8 n ≥ 2 – –8 1 8 – n ≤ –8 2

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EXAMPLE 2 Solving an Inequality Using Division Original inequality Divide each side by –3. Reverse inequality symbol. Simplify. –5 < m 15 > –3m 15 –3 < –3m –3

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EXAMPLE 3 Using the Division Property of Inequality Biology About 15,000 fruit-eating bats live on Barro Colorado Island. Yearly they eat up to 61,440,000 grams of fruit. Write and solve an inequality to find about how many grams g of fruit each bat eats yearly. SOLUTION

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EXAMPLE 3 Using the Division Property of Inequality Write an algebraic model. Divide each side by 15,000. Simplify. g ≤ 4096 15,000g ≤ 61,440,000 15,000g 15,000 ≤ 61,440,000 15,000 Each bat eats up to 4096 grams of fruit in a year. ANSWER

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GUIDED PRACTICE for Examples 1, 2, and 3 Solve the inequality. 1. t 6 > 4 t >24 6 t 6 > 4 6 Multiply each side by 6. Simplify

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GUIDED PRACTICE for Examples 1, 2, and 3 Solve the inequality. 2. 1 2 – x ≤ 10 x> –20 Multiply both sides by -2.

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GUIDED PRACTICE for Examples 1, 2, and 3 Solve the inequality. 3. 27 > –3t t < –9 Divide both sides by -3

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GUIDED PRACTICE for Examples 1, 2, and 3 Solve the inequality. 4. 9n < 63 7 < n Divide both sides by 9

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GUIDED PRACTICE for Examples 1, 2, and 3 5. Fruit Bats A bat that weighs about 25 grams can eat up to 2.5 times its body mass in figs in one night. How many grams g of figs can it eat? A bat can eat up to 62.5 g of figs. ANSWER

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GUIDED PRACTICE for Examples 1, 2, and 3 6. Baseball If you are at-bat 250 times this baseball season, how many hits must you get to have a batting average of at least 0.452? Let h represent the number of hits. Write a verbal model. Solution: hits At bats ≥ Target batting average h 250 ≥ 0.452 250 h ≥0.452 250 h≥113 Answer: You will have to get at least 113 hits to achieve the batting average of at least 0.452.

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GUIDED PRACTICE Solve the inequality. 8n > 32 u6u6 ≥ 3 -6s ≤ 54 -16k ≥ 96

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Example 1 Solving and Graphing a Two-Step Inequality 4y4y+10 < 18 Original inequality 4y4y < 8 Simplify. < –– Subtract 10 from each side. 18 10 4y4y+ (Subtraction.

Example 1 Solving and Graphing a Two-Step Inequality 4y4y+10 < 18 Original inequality 4y4y < 8 Simplify. < –– Subtract 10 from each side. 18 10 4y4y+ (Subtraction.

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