1. 6.5 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Solve Absolute Value Equations

2. 6.5 Warm-Up ANSWER 12, 12 ANSWER 1 2. Evaluate |x| – 2 when x = –3. 1. For a = –12, find, –a and |a|. The change in elevation as a diver explored a reef was –0.5 foot, 1.5 feet, –2.5 feet, and 2.25 feet. Which change in elevation had the greatest absolute value? ANSWER –2.5 ft 3.

3. 6.5 Example 1 SOLUTION The distance between x and 0 is 7. So, x = 7 or x = –7. ANSWER The solutions are 7 and –7. Solve x = 7.

4. 6.5 Guided Practice Solve (a) |x| = 3 and (b) |x| = 15. ANSWER 3, –3 a. 15, –15 b.

5. 6.5 Example 2 SOLUTION Rewrite the absolute value equation as two equations. Then solve each equation separately. x – 3 = 8 Write original equation. x – 3 = 8 or x – 3 = –8 Rewrite as two equations. x = 11 or x = –5 Add 3 to each side. ANSWER The solutions are 11 and –5. Check your solutions. x – 3 = 8. Solve

6. 6.5 Example 2 |x – 3| = 8 CHECK Substitute for x. Subtract. Simplify. The solution checks. |11 – 3| = 8 |–5 – 3| = 8 ?? | 8| = 8 |–8| = 8 ? ? Write original inequality. 8 = 8 8 = 8

7. 6.5 Example 3 SOLUTION First, rewrite the equation in the form ax + b = c. 3 2x – 7 – 5 = 4 3 2x – 7 = 9 2x – 7 = 3 Write original equation. Add 5 to each side. Divide each side by 3. 3 2x – 7 – 5 = 4. Solve

8. 6.5 Example 3 Next, solve the absolute value equation. 2x – 7 = 3 2x – 7 = 3 or 2x – 7 = –3 2x = 10 or 2x = 4 x = 5 or x = 2 Write absolute value equation. Rewrite as two equations. Add 7 to each side. Divide each side by 2. ANSWER The solutions are 5 and 2.

9. 6.5 Guided Practice Solve the equation. r – 7 = 9 2. 16, –2 ANSWER 2 s + 4.1 = 18.9 3. 7.4, –7.4 ANSWER 4 t + 9 – 5 = 19 4. –3, –15 ANSWER

10. 6.5 Example 4 3x + 5 + 6 = –2 Write original equation. 3x + 5 = –8 Subtract 6 from each side. ANSWER The absolute value of a number is never negative. So, there are no solutions. 3x + 5 + 6 = –2, if possible.Solve

11. 6.5 Guided Practice ANSWER no solution 5. 2 m – 5 + 4 = 2 Solve the equation, if possible 6. –3 n +2 –7 = –10 ANSWER  1,  3

12. 6.5 Absolute Deviation The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value: Absolute Deviation =

13. 6.5 Example 5 BASKETBALLS Before the start of a professional basketball game, a basketball must be inflated to an air pressure of 8 pounds per square inch (psi) with an absolute error of 0.5 psi. Absolute error is the absolute deviation of a measured value from an accepted value. Find the minimum and maximum acceptable air pressures for the basketball.

14. 6.5 Example 5 SOLUTION Let p be the air pressure ( in psi ) of a basketball. Write a verbal model. Then write and solve an absolute value equation. 0.5=p–8

15. 6.5 Example 5 p – 8 0.5 = p 8 – or p 8 –0.5 = – p 8.5 = or p 7.5 = Write original equation. Rewrite as two equations. Add 8 to each side. ANSWER The minimum and maximum acceptable pressures are 7.5 psi and 8.5 psi.

16. 6.5 Guided Practice 7. A volleyball league is preparing a two minute radio ad to announce tryouts. The ad has an absolute deviation of 0.05 minute. Find the minimum and the maximum acceptable times the radio ad can run. Minimum: 1.95 min Maximum: 2.05 min ANSWER

17. 6.5 Lesson Quiz Solve the equation, if possible. ANSWER –9, 17 ANSWERno solutions 1. 3| x – 4 | = 39 2. | x + 2 | + 7 = 3 ANSWER –5, 11 ANSWERno solutions 3. | 2x – 6 | – 18 = – 2 4. –2 | x – 5 | + 7 = 12

18. 6.5 Lesson Quiz A pattern for a 26 -inch skirt has an absolute deviation of 1.5 inches. Find the minimum and maximum skirt lengths that can be made from the pattern. 5. ANSWER minimum : 24.5 in.; maximum 27.5 in.