1. Section 4.4 Solving Absolute Value Equations and Inequalities

2. 4.4 Lecture Guide: Solving Absolute Value Equations and Inequalities Objective 1: Use absolute value notation to represent intervals.

3. AlgebraicallyVerbally The distance from 0 to x is d units. The distance from 0 to x is __________ than d units. The distance from 0 to x is __________ than d units. Numerical Example Graphical Example if or if and if or −3−3 3 0 ( ) −3−3 3 0 ( ) −3−3 3 0 Absolute Value Expressions For any real number x and any nonnegative real number d:

4. Use interval notation to represent the real numbers that are solutions of these inequalities. 1.2.

5. Write an absolute-value inequality to represent each set of points. 3. 4.

6. Write an absolute-value inequality to represent each set of points. 5. The points between – 6 and 6. 6. −2−20−4−4−6−6−8−8−10246810

7. Use absolute value to represent the distance between these real numbers. 7. 8. 9. y and x x and 10 x and

8. 10. Write an absolute value equation indicating that the distance from to a isunits.

9. Objective 2: Solve absolute value equations and inequalities.

10. AlgebraicallyVerbally x is either d units _________ or __________ of a. x is _________ than d units from a. Equivalent Expressions Graphically a – da + d a ( ) a ( ) a Solving Absolute Value Equations and Inequalities For any real number x and any nonnegative real number d: a – da + d a – da + d

11. Similar statements can also be made about the order relations less than or equal to and greater than or equal to Expressions with d negative are examined in the group exercises at the end of this section.

12. Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 11.

13. Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 12.

14. Solve each equation or inequality algebraically. 13.

15. Solve each equation or inequality algebraically. 14.

16. Solve each equation or inequality algebraically. 15.

17. Solve each equation or inequality algebraically. 16.

18. Solve each equation or inequality algebraically. 17.

19. Solve each equation or inequality algebraically. 18.

20. Solve each equation or inequality algebraically. 19.

21. Solve each equation or inequality algebraically. 20.

22. If, then a and b are equal in magnitude but their signs can either agree or disagree. Thus is equivalent to ______________ or ______________. Use this result to solve the next two problems.

23. 21. Solve.

24. 22. Solve.

25. 23. Use the graph to solve each equation or inequality. (a) (b) (c)

26. 24. Use the table to solve each equation or inequality. (a) (b) (c)

27. Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 25.

28. Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 26.

29. Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 27.

30. Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 28.

31. Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 29.

32. Solve each equation or inequality algebraically. Use the given graph to solve each equation or inequality graphically. Then complete the table to check your solutions numerically. 30.

33. Write an absolute value inequality to represent the following intervals. Hint: First sketch a graph of these inequalities. 31. 32.

34. Write an absolute value inequality to represent the following intervals. Hint: First sketch a graph of these inequalities. 33.34.

35. 35. The correct torque setting for the lug bolts on a race car is 85 foot-pounds with a tolerance of ±3 foot-pounds. (a) Express the acceptable torque setting as an absolute value inequality. (b) Express the acceptable torque setting as a compound linear inequality.

36. 35. The correct torque setting for the lug bolts on a race car is 85 foot-pounds with a tolerance of ±3 foot-pounds. (c) Determine the lower and upper limits of the interval.