Absolute Value Equations

1. Absolute Value Equations

2. Objective • I will be able to solve absolute value equations.

3. -2 -1 0 1 2 3 Absolute Value • Absolute value of a number is its distance from zero on a number line. 2 units

4. Solving Equations of the Form

5. Example 1 • Solve • Since 2 is positive is equivalent to p = 2 or p = -2 • To check, let p = 2 and then p = -2 in the original equation. Original Equation Original Equation Let p = -2 Let p = 2 2 = 2 True 2 = 2 True

6. Solution • The solutions are 2 and -2 or the solution set is {2, -2}

7. Give it a try!  • Solve

8. Example 2 • Solve Translate: 5 w + 3 = 7 OR 5 w + 3 = -7 Solve both equations for w 5 w + 3 = 7 5 w + 3 = -7 5w = 4 5 w = -10 w = w = -2

9. Solution • {-2, } Check your solution, let w = -2 then let w = -4/5

10. Give it a try!  • Solve

11. Example 3 • Solve 24 and -20

12. Give it a try!  • Solve

13. Example 4: Isolate the absolute value expression! • Solve Subtract 5 from both sides 2x = 2 2x = -2 x = 1 x = -1 The solutions are -1 and 1

14. Give it a try!  • Solve

15. Example 5: ZERO • Solve • We are looking for all numbers whose distance from 0 is zero units. The only number is 0. The solution is 0.

16. Example 6 • Solve: Subtract 25 from both sides Divide both sides by 2 The absolute value of a number is NEVER negative, so this equation has no solution!

17. Give it a try!  • Solve:

18. Example 7 • Solve: The absolute value of any expression is never negative, so no solution exists!

19. When are absolute value expressions equal? Same Same Opposites Opposites Two absolute value expressions are equal when the expressions inside the absolute value bars are equal to or are opposites of teach other.

20. Example 8: • Solve: This equation is true if the expressions inside the absolute value bars are equal to or are opposites of each other. 3x + 2 = 5x – 8 OR 3x + 2 = -(5x -8)

21. Solve each equation 3x + 2 = 5x – 8 OR 3x + 2 = -(5x – 8) -2x + 2 = -8 3x + 2 = -5x + 8 -2x = 10 8 x = 6 x = 5 x = ¾ The solutions are ¾ and 5.

22. Give it a try!  • Solve:

23. Example 9 • Solve: x – 3 = 5-x OR x – 3 = -(5-x) 2 x – 3 = 5 x – 3 = -5 + x 2x = 8 x-x – 3 = - 5 x = 4 0 – 3 = -5 -3 = -5 False

24. Solution to Example 9 • The equation on the right simplified to a false statement. So the only solution to this equation is 4.

25. Give it a try!  • Solve