1. Section 4.4 Solving Absolute Value Equations and Inequalities

2. 4.4 Lecture Guide: Solving Absolute Value Equations and Inequalities Objective: Solve absolute value equations and inequalities algebraically.

3. Algebraically Verbally Algebraic Example Graphical Example Absolute Value Expressions For any real number x and any nonnegative real number d: The distance from 0 to x is d units. ifor −3−3 3 0

4. Algebraically Verbally Algebraic Example Graphical Example Absolute Value Expressions For any real number x and any nonnegative real number d: The distance from 0 to x is less than d units. if and ( ) −3−3 3 0

5. Algebraically Verbally Algebraic Example Graphical Example Absolute Value Expressions For any real number x and any nonnegative real number d: The distance from 0 to x is more than d units. if or ( ) −3−3 3 0

6. Use interval notation to represent the real numbers that are solutions of these inequalities. 1. 2.

7. Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 3. 4. −2−20−4−4−6−6−8−8−10246810−2−20−4−4−6−6−8−8−10246810

8. Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 5. 6. The points between and 6. −2−20−4−4−6−6−8−8−10246810−2−20−4−4−6−6−8−8−10246810

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

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

11. Absolute Value Expression Verbally Graphically Equivalent Expression Solving Absolute Value Equations and Inequalities For any real numbers x and a and positive real number d: x is either d units left or right of a. a − d a a + d

12. Absolute Value Expression Verbally Graphically Equivalent Expression Solving Absolute Value Equations and Inequalities For any real numbers x and a and positive real number d: x is less than d units from a. ( ) a − d a a + d

13. Absolute Value Expression Verbally Graphically Equivalent Expression Solving Absolute Value Equations and Inequalities For any real numbers x and a and positive real number d: x is more than d units from a. a − d a a + d ( )

14. 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.

15. 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. 12. −2−20−4−4−6−6−8−8−10246810−2−20−4−4−6−6−8−8−10246810

16. Write an absolute-value inequality to represent each interval. First graph the interval and use this graph to assist you in writing the inequality. 13. 14. The numbers that are at least –1 and at most 5. The numbers that are less than –8 or greater than 0. −2−20−4−4−6−6−8−8−10246810−2−20−4−4−6−6−8−8−10246810

17. Solve each equation or inequality algebraically. 15.

18. Solve each equation or inequality algebraically. 16.

19. Solve each equation or inequality algebraically. 17.

20. Solve each equation or inequality algebraically. 18.

21. Solve each equation or inequality algebraically. 19.

22. Solve each equation or inequality algebraically. 20.

23. Solve each equation or inequality algebraically. 21.

24. Solve each equation or inequality algebraically. 22.

25. Objective: Solve absolute value equations and inequalities using tables and graphs.

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

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

28. Solve each equation or inequality algebraically. Then use a calculator to generate a graph and a table to verify your solutions. See Calculator Perspective 4.4.1. 25.

29. Solve each equation or inequality algebraically. Then use a calculator to generate a graph and a table to verify your solutions. See Calculator Perspective 4.4.1. 26.

30. Solve each equation or inequality algebraically. Then use a calculator to generate a graph and a table to verify your solutions. See Calculator Perspective 4.4.1. 27.

31. 28. Solve each equation or inequality algebraically. Then use a calculator to generate a graph and a table to verify your solutions. See Calculator Perspective 4.4.1.

32. 29. Solve each equation or inequality algebraically. Then use a calculator to generate a graph and a table to verify your solutions. See Calculator Perspective 4.4.1.

33. 30. Solve each equation or inequality algebraically. Then use a calculator to generate a graph and a table to verify your solutions. See Calculator Perspective 4.4.1.

34. Ifthenandare equal in magnitude but their signs can either agree or disagree.Thus is equivalent to 31. Solve ______________ or ______________. Use this result to solve the next two problems.

35. 32. Solve Ifthenandare 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.

36. Write an absolute value inequality to represent the following intervals. 33.

37. Write an absolute value inequality to represent the following intervals. 34.

38. Write an absolute value inequality to represent the following intervals. 35.

39. Write an absolute value inequality to represent the following intervals. 36.

40. 37. 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. (c) Determine the lower and upper limits of the interval.