1. Solving Absolute Value Equations and Inequalities
Lesson 1 – 6
Solving Absolute Value Equations and Inequalities

2. Objectives Evaluate expressions involving absolute values.
Solve absolute value equations and inequalities.

3. Absolute Value Expressions
The absolute value of a number is its distance from 0 on the number line.
Since distance is nonnegative, the absolute value of a number is always positive.
The symbol |x| is used to represent the absolute value of a number x.

4. Evaluating Expressions
When evaluating expressions that contain absolute values, the absolute value bars act as a grouping symbol.
Perform any operations inside the absolute value bars first.

5. Example 1 Evaluate if y = -3. Fill in -3 for y.
Simplify inside the absolute value bars.

6. Example 2 Solve the following equation. Add 18 to each side.
When solving absolute value equations, you MUST check both solutions. In this case, both solutions work.

7. Calculator Check

8. Example 3 Solve the following equation.
The absolute value must be on one side of the equation by itself. Therefore, we must subtract 9 from each side.
The solution to an absolute value can never be negative. Therefore, this equation has no solution.
This is the symbol for the empty set.

9. Example 4 Remember to check both solutions.
This time one of them does not work.
Example 4
Subtract x from each side.
Distribute the -1.
Add 3x to each side.
Add 2 to each side.
Subtract 6 from each side.
Divide each side by 2.

10. Calculator Check

11. Example 5 Divide each side by 7. Now set up the two equations.
Add 13 to each side.
Divide each side by 4.
Remember to check both solutions.

12. Calculator Check

13. Example 6
Divide each side by 3.
Check each answer!

14. Absolute Value Inequality (<)
Absolute value inequalities which involve less than ( < ) or less than or equal to ( ≤ ) use the word and.
For example, how would we write:
Or, we can write
Both ways are correct, but I prefer the 2nd one.

15. Absolute Value Inequality (>)
Absolute value inequalities involving greater than ( > ) or greater than or equal to ( ≥ ) use the word or.
For example, consider:
This is the only option for writing this one besides interval notation.

16. Example 7
Do we use “and” or “or”? OR 2 6

17. Example 8
AND -3 -2 1

18. SOL Problem