1. EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c A professional baseball should weigh 5.125 ounces, with a tolerance of 0.125 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

2. EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c STEP 2Solve the inequality. Write inequality. Write equivalent compound inequality. Add 5.125 to each expression. |w – 5.125| ≤ 0.125 – 0.125 ≤ w – 5.125 ≤ 0.125 5 ≤ w ≤ 5.25 So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below. ANSWER

3. EXAMPLE 6 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. Write a range as an absolute value inequality

4. EXAMPLE 6 Mean of extremes = = 7.875 7.5 + 8.25 2 Find the tolerance by subtracting the mean from the upper extreme. STEP 2 Tolerance = 8.25 – 7.875 Write a range as an absolute value inequality = 0.375

5. EXAMPLE 6 STEP 3 Write a verbal model. Then write an inequality. A mat is acceptable if its thickness t satisfies |t – 7.875| ≤ 0.375. ANSWER Write a range as an absolute value inequality

6. GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. 10. |x + 2| < 6 The absolute value inequality is equivalent to x + 2 – 6 First Inequality Second Inequality x + 2 < 6x + 2 > – 6 x < 4x > – 8 Write inequalities. Subtract 2 from each side. SOLUTION

7. GUIDED PRACTICE for Examples 5 and 6 ANSWER The solutions are all real numbers less than – 8 or greater than 4. The graph is shown below.

8. GUIDED PRACTICE for Examples 5 and 6 Solve the inequality. Then graph the solution. 11. |2x + 1| ≤ 9 The absolute value inequality is equivalent to 2x + 1 – 9 First Inequality Second Inequality 2x < 82x > – 10 Write inequalities. Subtract 1 from each side. SOLUTION 2x + 1 > – 9 2x + 1 < 9 x < 4x > – 5 Divide each side 2

9. GUIDED PRACTICE for Examples 5 and 6 ANSWER The solutions are all real numbers less than – 5 or greater than 4. The graph is shown below.

10. GUIDED PRACTICE for Examples 5 and 6 12. |7 – x| ≤ 4 Solve the inequality. Then graph the solution. The absolute value inequality is equivalent to 7 – x – 4 First Inequality Second Inequality – x < – 3– x > – 11 Write inequalities. Subtract 7 from each side. SOLUTION x < 3x > 11 Divide each side (– ) sign 7 – x > – 4 7 – x < 4

11. GUIDED PRACTICE for Examples 5 and 6 ANSWER The solutions are all real numbers less than 3 or greater than 11. The graph is shown below.

12. GUIDED PRACTICE for Examples 5 and 6 13. Gymnastics: For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses. SOLUTION STEP 1 Calculate the mean of the extreme mat thicknesses. Mean of extremes = = 7.875 7.5+ 8.25 2 Find the tolerance by subtracting the mean from the upper extreme. STEP 2 Tolerance = 8.25 – 7.875 = 0.375

13. GUIDED PRACTICE for Examples 5 and 6 STEP 3 A mat is unacceptable if its thickness t satisfies |t – 7.875| > 0.375. ANSWER