1. Absolute Value Equations & Inequalities
EQ: How does solving an absolute value equation/inequality compare to solving a linear equation/inequality?

2. I am 3 “miles” from my house
|B| = 3
B = 3 or B = -3

3. Review Ex 1) |7| = Ex 2) |-5| = Ex 3) 5|2 – 4| + 2 =
Absolute Value: the distance a number is from zero (always positive)
Ex 1) |7| =
Ex 2) |-5| =
Ex 3) 5|2 – 4| + 2 =

4. Absolute Value Equations
Ex 1) Solve: |x| = 8
Ex 2) Solve: |x| = 25
Ex 3) Solve: |x| = -10

5. Have two solutions because the absolute value of a positive number is the same as the absolute value of a negative number.

6. Solving Absolute Value Equations
Step 1: Get “bars” alone on one side Step 2: Re-write as two equations; flip the signs on the RIGHT side (drop the bars) Step 3: Solve both equations Step 4: Check for an extraneous solution!!

7. Extraneous Solution
A value you get after correctly solving the problem that does not actually satisfy the equation.

8. Ex 4)
3|x + 2| - 7 = 14
3|x + 2| = 21
|x + 2| = 7

9. |x + 2| = 7
x + 2 = x + 2 = -7
x = 5 or x = -9

10. Check Your Answers:
x = 5 x = -9
3|x + 2| - 7 = 14
3|x + 2| - 7 = 14
3|5 + 2| - 7 = |-9 + 2| - 7 = 14
3|7| - 7 = |-7| - 7 = 14
14 = = 14

11. Ex 5)
|3x + 2| = 4x + 5
3x + 2 = 4x + 5
3x + 2 = -4x – 5

12. 3x + 2 = -4x – 5
3x + 2 = 4x + 5

13. Check Your Solutions
|3x + 2| = 4x + 5

14. |x – 4| + 7 = 2 |x – 4| = -5 NO SOLUTION
Ex 6)
|x – 4| + 7 = 2 |x – 4| = -5 NO SOLUTION

15. Solving by Graphing
Step 1: Enter the left & right side into Y1 & Y2 PRESS → NUM #1 abs( Step 2: Find the first intersection Step 3: Find the second intersection if there is one
MATH #5
2nd
TRACE

16. 3|x + 2| -7 = 14

17. |3x + 2| = 4x + 5

18. Solve: |5x| + 10 = 55 Solve: |x – 3| = 10 Solve: 2|y + 6| = 8
Practice
Solve: |5x| + 10 = 55
Solve: |x – 3| = 10
Solve: 2|y + 6| = 8
Solve: |a – 5| + 3 = 2
Solve: |4x + 9| = 5x + 18

19. Thank about it…
If |x| < 5 what are some possible values of x?
If |x| > 5 what are some possible values of x?

20. Solving Absolute Value Inequalities
  Since the inequalities are absolute value, there are still going to be two solutions
When writing the second equation, be sure to flip the inequality sign.
Also, when dividing by a negative the inequality sign flips.

21. Ex 1)

22. Ex 2)

23. You Try! Solve | 2x + 3 | < 6

24. Ex 3) Solve

25. You Try! Solve |2x-5|>x+1.

26. Quick Write:
Think of some objects or situations that need to be within a certain “value” – if you go too much over it would be a bad thing, and if you go too much under it would also be a bad thing:

27. Tolerance There are strict height requirements to be a “Rockette”
You must be between 66 inches 70.5 inches

28. Tolerance:
The difference between a desired measurement and its maximum and minimum allowable values.
Most
Allowed
Least
Allowed
Perfect
Amount
Tolerance
Tolerance

29. Tolerance: __________
Ex 1) The doctor says that you need to stay between 125 and 135 lbs to be at a healthy weight.
Min: _______ Perfect: _______ Max: ________
Tolerance: __________

30. Example 2)
Workers at a hardware store take their morning break no earlier than 10 am and no later than noon. Let c represent the time the workers take their break. Write an absolute value inequality to represent the situation.

39. Exit Ticket:
How is solving an absolute value inequality different than solving an absolute value equation?
What is the difference between the solutions of an equation and the solutions of an inequality?