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EXAMPLE 5

Baseball. 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. STEP 1. Write a verbal model. Then write an inequality. EXAMPLE 5.

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EXAMPLE 5

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  1. Baseball 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. STEP1 Write a verbal model. Then write an inequality. EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c SOLUTION

  2. STEP2 Solve the inequality. ANSWER So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below. EXAMPLE 5 Solve an inequality of the form |ax + b| ≤ c |w – 5.125| ≤ 0.125 Write inequality. Write equivalent compound inequality. – 0.125 ≤ w – 5.125 ≤ 0.125 5 ≤ w ≤ 5.25 Add 5.125 to each expression.

  3. Gymnastics 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. Calculate the mean of the extreme mat thicknesses. STEP1 EXAMPLE 6 Write a range as an absolute value inequality SOLUTION

  4. Mean of extremes ==7.875 STEP2 Find the tolerance by subtracting the mean from the upper extreme. 7.5 + 8.25 2 EXAMPLE 6 Write a range as an absolute value inequality Tolerance = 8.25 –7.875 =0.375

  5. STEP3 Write a verbal model. Then write an inequality. ANSWER A mat is acceptable if its thickness tsatisfies |t – 7.875| ≤ 0.375. EXAMPLE 6 Write a range as an absolute value inequality

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

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

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

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

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

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

  12. Mean of extremes ==7.875 Calculate the mean of the extreme mat thicknesses. STEP1 7.5+ 8.25 2 STEP2 Find the tolerance by subtracting the mean from the upper extreme. for Examples 5 and 6 GUIDED PRACTICE 13.Gymnastics: For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses. SOLUTION =0.375 Tolerance = 8.25 –7.875

  13. ANSWER A mat is unacceptable if its thickness tsatisfies |t – 7.875| > 0.375. for Examples 5 and 6 GUIDED PRACTICE STEP3

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