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EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c A professional baseball should weigh ounces, with a tolerance of ounce. Write and solve an absolute value inequality that describes the acceptable weights for a baseball. Baseball SOLUTION Write a verbal model. Then write an inequality. STEP 1

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**Solve an inequality of the form |ax + b| ≤ c**

EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c STEP 2 Solve the inequality. |w – 5.125| ≤ Write inequality. Write equivalent compound inequality. – ≤ w – ≤ 5 ≤ w ≤ 5.25 Add to each expression. So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below. ANSWER

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EXAMPLE 6 Write a range as an absolute value inequality The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 inches and 8.25 inches, inclusive. Write an absolute value inequality describing the acceptable mat thicknesses. Gymnastics SOLUTION STEP 1 Calculate the mean of the extreme mat thicknesses.

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EXAMPLE 6 Write a range as an absolute value inequality Mean of extremes = = 7.875 2 Find the tolerance by subtracting the mean from the upper extreme. STEP 2 Tolerance = 8.25 – 7.875 = 0.375

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EXAMPLE 6 Write a range as an absolute value inequality STEP 3 Write a verbal model. Then write an inequality. A mat is acceptable if its thickness t satisfies |t – 7.875| ≤ ANSWER

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GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. |x + 2| < 6 ANSWER –8 < x < 4 The solutions are all real numbers less than – 8 or greater than 4. The graph is shown below.

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GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. 11. |2x + 1| ≤ 9 ANSWER –5 ≤ x ≤ 4 The solutions are all real numbers less than –5 or greater than 4. The graph is shown below.

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GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. |7 – x| ≤ 4 3 ≤ x ≤ 11 ANSWER The solutions are all real numbers less than 3 or greater than 11. The graph is shown below.

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GUIDED PRACTICE for Examples 5 and 6 Gymnastics: For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses. A mat is unacceptable if its thickness t satisfies |t – 7.875| > ANSWER

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