Notes for absolute value equations and inequalities missed because of a field trip. Please let me know if you find a mistake!

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 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