The document provides an overview of absolute value functions and how to solve absolute value equations and inequalities. It defines absolute value, discusses evaluating absolute value expressions, and solves absolute value equations by isolating the absolute value. It also explains how to solve absolute value inequalities by considering whether the absolute value is less than, greater than, or equal to the right side of the inequality and interpreting the solutions. Examples of each type of problem are worked out step-by-step.

1. For those of you who missed it... Absolute Value!!!

2. 1. 2. 3. 4. 5. 6. Domain: (- ∞ , ∞ ) Range: [2, ∞ ) Domain: (- ∞ , ∞ ) Range: (- ∞, -2] Domain: (- ∞ , ∞ ) Range: [-1, ∞ ) Domain: (- ∞ , ∞ ) Range: [-2, ∞ ) Domain: (- ∞ , ∞ ) Range: (- ∞, 3] Domain: (- ∞ , ∞ ) Range: (- ∞, 1 ] Answers to Absolute Value Worksheet

3. Answers to Absolute Value Worksheet f(x) = 2|x - 3| + 3 f(x) = 1/3|x + 5| + 3

4. Answers to Absolute Value Worksheet f(x) = -3/2|x - 5| - 4 f(x) = -3/2|x + 6| - 2

5. Answers to Absolute Value Worksheet f(x) = 3|x - 4| - 10 f(x) = -2|x - 4| + 9

6. Answers to Absolute Value Worksheet f(x) = 4|x + 5| + 9 f(x) = 3/5|x + 4| - 8

7. 2x + 3 = 6 2x + 3 = -6 2x = 3 2x = -9 x = 3/2 x = -9/2 Solving Absolute Value Equations... Absolute Value: For any real number x, |x| = { -x, if x < 0 0, if x = 0 x, if x > 0 Recall: When solving equations, isolate the absolute value. Here are a few examples... 1. 5|2x + 3| = 30 |2x + 3| = 6 Don't forget to check!!! 5|6| = 30  5|-6| = 30  solution set: {3/2, -9/2}

8. example 2: -2|x + 2| + 12 = 0 -2|x + 2| = -12 |x + 2| = 6 isolate the absolute value! x + 2 = -6 x = -8 x + 2 = 6 x = 4 -2|-6| + 12 = 0 -2|6| + 12 = 0 {4, -8}

9. 5|3×+ 7|=-65 |3x + 7|=-13 absolute value cannot be negative!! example 3: {}

10. {3} example 4: |2x + 12| = 7x - 3 2x + 12 = 7x - 3 2x + 15 = 7x 15 = 5x 3 = x 2x + 12 = -(7x - 3) 2x + 12 = -7x + 3 9x + 12 = 3 9x = -9 x = -1 |18| = 18 |10| = -10 reject!

11. Absolute Value Inequalities Recall: |ax+b|=c, where c>0 ax+b=c ax+b=-c |ax+b|<c think: between &quot;and&quot; -c < ax+b < c ax+b < c and ax+b > -c ax+b>c or ax+b<-c why? we will express < or ≤ as an equivalent conjunction using the word AND |ax+b|>c think: beyond &quot;or&quot; we will express > or ≥ as an equivalent disjunction using the word OR

12. I. Less than... a) |x| < 5 x < 5 and x >-5 written as solution set: {x: -5< x < 5} Graph on a number line! use open circles! shade between!!! 1 0 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

13. b) |2x - 1| < 11 2x-1<11 and 2x-1>-11 2x < 12 and 2x > -10 x < 6 and x > -5 {x: -5 < x < 6} 1 0 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

14. c) 4|2x + 3| - 11 ≤ 5 4|2x + 3| ≤ 16 |2x + 3| ≤ 4 2x + 3 ≤ 4 AND 2x + 3 ≥ -4 2x ≤ 1 AND 2x ≥ -7 x ≤ 1/2 AND x ≥ -7/2 notice closed ends! -1 0 -2 -3 -4 -5 1 2 3 4 5

15. d) |7x + 10| < 0 think.... can an absolute value be negative??? NO!! {}

16. II. Greater than... a) |x| > 5 x > 5 or x < -5 written as solution set: {x: x > 5 or x < -5} I nterval notation (we will not use this, just set, but as an FYI): (- ∞ , -5) ∪ (5, ∞ ) Graph on a number line! use open circles! shade beyond!!! 1 0 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

17. b) |2x - 1| > 11 2x-1>11 or 2x-1<-11 2x > 12 or 2x < -10 x > 6 or x < -5 {x: x > 6 or x < -5} 1 0 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

18. c) 4|2x + 3| - 11 ≥ 5 4|2x + 3| ≥ 16 |2x + 3| ≥ 4 2x + 3 ≥ 4 OR 2x + 3 ≤ -4 2x ≥ 1 OR 2x ≤ -7 x ≥ 1/2 OR x ≤ -7/2 notice closed ends! -1 0 -2 -3 -4 -5 1 2 3 4 5

19. d) |7x + 10| ≥ 0 think.... when is an absolute value greater than or equal to 0??? always!! {x: x ∈ R } x is a real number! -1 0 -2 -3 -4 -5 1 2 3 4 5

20. LAST ONE! 5 < |x + 3| ≤ 7 |x + 3| >5 |x + 3| ≤ 7 x + 3 > 5 or x + 3 < -5 x+ 3 ≤ 7 and x + 3 ≥ -7 x > 2 or x < -8 x ≤ 4 and x ≥ -10 now graph it! graph above the number line and look for the overlap. This is where your solution will appear. {x: -10 ≤ x < 8 or 2 < x ≤ 4} 1 0 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

21. Remember to see me, email me or ask on the wiki if you have questions!! -Ms. P