1. Solving Absolute Value Inequalities

2. Definition of Absolute Value
Recall that the definition of the absolute value of a number 𝑎 is:
𝑎 = 𝑎, if 𝑎≥0 −𝑎, if 𝑎<0
For this lesson you will learn to solve absolute value inequalities, and we can use the definition above to understand what these are
So, suppose that 𝑥 is an unknown number and that 𝑏 is positive or zero
We want to know how to solve 𝑥 >𝑏

3. Absolute Value Inequalities
By our definition,
𝑥 >𝑏⟹ 𝑥>𝑏, if 𝑥≥0 −𝑥>𝑏, if 𝑥<0
Note that this is really saying the following:
𝑥>𝑏 AND 𝑥≥0 OR
𝑥<−𝑏 AND 𝑥<0

4. 𝑥 >𝑏

5. 𝑥 >𝑏

6. 𝑥 >𝑏
From the two previous slides we see that 𝑥>𝑏 AND 𝑥≥0 OR 𝑥<−𝑏 AND 𝑥<0 is the same as 𝑥<−𝑏 OR 𝑥>𝑏 As a graph:

7. 𝑥 <𝑏 Let’s use the same method to determine the solution to 𝑥 <𝑏
Using the definition of absolute value we have
𝑥 <𝑏⟹ 𝑥<𝑏, if 𝑥≥0 −𝑥<𝑏, if 𝑥<0
This means the same as
𝑥<𝑏 AND 𝑥≥0 OR
𝑥>−𝑏 AND 𝑥<0

8. 𝑥 <𝑏

9. 𝑥 <𝑏

10. 𝑥 <𝑏
Now, we have the following −𝑏<𝑥<0 OR 0≤𝑥<𝑏 But a disjunction (“OR”) joins everything into one set. So we can simplify the following to −𝑏<𝑥<𝑏 As a graph:

11. Absolute Value Inequalities
We have shown the following: Suppose that 𝑏≥0. Then:
*If |𝑥|>𝑏, then 𝑥<−𝑏 OR 𝑥>𝑏
*If 𝑥 <𝑏, then−𝑏<𝑥<𝑏
We will use these to solve absolute value inequalities

12. Absolute Value Inequalities
Example: Solve the absolute value inequality. Record the result in interval notation. 𝑥+2 >5

13. Guided Practice
Solve the following absolute value inequalities. Record your answer as a graph and in interval notation.
𝑥−3 ≥8
1 2 3𝑥+1 >5
2⋅ 4−𝑥 −5≥1
− 3𝑏+1 −5<−9

14. Absolute Value Inequalities
Example: Solve the absolute value inequality. Record the result in interval notation. 𝑥−7 ≤9

15. Guided Practice
Solve the following absolute value inequalities. Record your answer as a graph and in interval notation.
2−3𝑎 <8
−2⋅ 3𝑥+1 +1>−9
3⋅ 𝑥+7 +3≤15
− 2𝑥−7 −1≥−15

16. Tolerance
Tolerance refers to the amount of error we are willing to tolerate in, for example, manufacturing a product
For example, MLB requires official baseballs to weigh ounces with a tolerance of ounces
We can represent this with the absolute value inequality actual weight−ideal weight ≤tolerance
For baseball weights, the formula is 𝑤−5.125 ≤0.125
What range of values 𝑤 may an official baseball have?

17. Concentrate!