1. 2.6 Solving absolute value inequalities

2. What we will learn
Solve absolute value inequalities
Story problem

3. Needed vocab
Absolute value inequality: inequality that contains an absolute value expression

4. Ex. 1 solving when absolute value by itself
Solve and graph:
𝑥+7 ≤2
𝑥+7≤2 𝑥+7≥−2
−7 − −7 −7
𝑥≤−5 𝑥≥−9
Steps
1. make a T
2. rewrite original with no changes
3. rewrite left, but flip the sign and change sign of answer
4. solve for variable
Note: Absolute value has to be greater than or equal to 0. If less than 0, then no solution

5. Your Practice 10−𝑚 ≥5 10−𝑚≥5 10−𝑚≤−5 −10 −10 −10 −10 −𝑚≥−5 −𝑚≤−15
10−𝑚≥5 10−𝑚≤−5
−10 −10 − −10
−𝑚≥− −𝑚≤−15
−𝑚 −1 ≥ −5 − −𝑚 −1 ≤ −15 −1
𝑚≤ 𝑚≥15 5 15

6. Ex. 2 when absolute value not by itself
4 2𝑥−5 +1>21
−1 −1
4 2𝑥−5 >20
4 2𝑥−5 4 > 20 4
2𝑥−5 >5
2𝑥−5> 𝑥−5<−5
2𝑥> 𝑥<0
2𝑥 2 > 𝑥 2 < 0 2
𝑥> 𝑥<0
Steps
1. Get by itself
2. make a T
3. rewrite original with no changes
4. rewrite left, but flip the sign and change sign of answer
5. solve for variable

7. Your practice 3 𝑑+1 −7≥−1 +7 +7 3 𝑑+1 ≥6 3 𝑑+1 3 ≥ 6 3 𝑑+1 ≥2
3 𝑑+1 ≥6
3 𝑑+1 3 ≥ 6 3
𝑑+1 ≥2
𝑑+1≥ 𝑑+1≤−2
−1 − −1 −1
𝑑≥ 𝑑≤−3

8. Ex. 3 absolute deviation story problem
You are buying a new computer. The table shows the prices of computers in a store advertisement. You are willing to pay the mean price with an absolute deviation of at most $ how many computer prices meet your condition?
𝑥−𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒 =𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Given value is mean
Mean = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑖𝑡𝑒𝑚𝑠 𝑡𝑜𝑡𝑎𝑙 𝑖𝑡𝑒𝑚𝑠
Mean =
Mean = 664
𝑥−664 ≤100
𝑥−664≤ 𝑥−664≥−100
𝑥≤ 𝑥≥564
six