1. 1.4 Absolute Value Equations Absolute value: The distance to zero on the number line. We use two short, vertical lines so that |x| means “the absolute value of x”

2. Absolute Value Example |3| = |-3| = 3 3 |a| = a

3. Absolute Value Example |a| = a This only works for a ≥ 0 This won’t work all of the time. Does it work for a = 1? Does it work for a = 0? Does it work for a = -1?

4. Absolute Value Example |a| = a if a ≥ 0 if a < 0 |a| = -a

5. Evaluate Absolute Value Evaluate |5-x 2 | for x = -3 |5-x 2 | |5-(-3) 2 | |5-9| |-4| 4 Substitute using x = -3 Simplify the exponent Subtraction Definition of Absolute Value Given

6. Evaluate Absolute Value Evaluate |x 2 -4x-6| for x = -1 |x 2 -4x-6| |(-1) 2 -4(-1)-6| |1-4(-1)-6| |-1| 1 Substitute using x = -1 Simplify the exponent Subtraction Definition of Absolute Value Given |5-6| Addition |1+4-6| Multiplication

7. Solving Absolute Value Equations Your biggest concern with solving is that there are typically 2 cases to solve! Solve: |x -1| = 5 For x -1 being positive, we can just throw the ||’s into the trash and continue. But what about the case where x -1 is negative?

8. Solving Absolute Value Equations Solve: |x -1| = 5 Case 1: x -1 = 5 x = 6 Case 2: x -1 = -5 x = -4 x = {-4, 6} There’s more than one answer. That means there is a set of answers. So we need to use { }’s around our set.

9. Solving Absolute Value Equations Solve: |2x -3| = 16 Case 1: 2x -3 = 16 x = 19 2 Case 2: 2x -3 = -16 2x = 192x = -13 x = -13 2 x = -13, 19 2 2 {}Oops!

10. Solving the impossible? Solve: |2x -3| +5 = 0 |2x -3| = -5 |something| is trying to be negative ???

11. Solving the impossible? Solve: |2x -3| +5 = 0 So, no, this problem doesn’t have a solution. x = { } This means the solution set is empty. x = ∅ Same thing, except fancier.

12. Why Should I Check It? So why do math teachers make such a big deal about checking your answers? Isn’t being careful while solving good enough? Sorry, no. Prepare to meet a most deceptive type of problem.

13. Why Should I Check It? Solve: |2x +8| = 4x -2 Case 1: 2x +8 = 4x - 2 5 = x Case 2: 10 = 2x 2x +8 = -(4x - 2) 6x = -6 2x +8 = -4x +2 x = -1 x = {-1,5}

14. Why Should I Check It? Check: |2x +8| = 4x -2; x = {-1,5} Check 5: |2(5) +8| = 4(5) -2 |18| = 18 Check -1: |10 +8| = 20 -2 18 = 18 |2(-1) +8| = 4(-1) -2 |6| = -6 |-2 +8| = -4 -2 6 = -6 Good answer. We’ll keep you. Aaaargh! That’s a bad answer!

15. Why Should I Check It? Edit: |2x +8| = 4x -2; x = {-1,5} x = {-1,5}This is wrong. x = 5This is right. -1 didn’t check, so it is rejected!

16. Why Should I Check It? After you finish tonight’s homework, for every equation that you didn’t check, mark it wrong so we can save time grading.