1. – 8 and 8 is a solution of the
Recall :
Absolute value | x | : is the distance between x and 0. If | x | = 8, then
– 8 and 8 is a solution of the
equation because both 8 and -8 are located a distance of 8 from 0.

2. Absolute Value (of x) Symbol lxl
The distance x is from 0 on the number line.
Always positive
Ex: l-3l=3
Recall:
You can solve some absolute-value equations using mental math. For instance, you learned that the equation
| x | 3 has two solutions: 3 and 3.
To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

3. Solving an Absolute-Value Equation:
Solve | x  2 |  5
Solve | 2x  7 |  5  4

4. Answer :: Solving an Absolute-Value Equation Solve | x  2 |  5
The expression x  2 can be equal to 5 or 5.
x  2 IS POSITIVE
| x  2 |  5
x  2  5
x  7
x  3
x  2 IS NEGATIVE
| x  2 |  5
x  2  5
The equation has two solutions: 7 and –3.
CHECK
| 7  2 |  | 5 |  5
| 3  2 |  | 5 |  5

5. Answer :: Solve | 2x  7 |  5  4 SOLUTION
Isolate the absolute value expression on one side of the equation.
| 2x  7 |  5  4
| 2x  7 |  9
2x  7 IS NEGATIVE
2x  7 IS POSITIVE
2x  7  +9
2x  7  9
2x  16
x  8
x  1
2x  2
TWO SOLUTIONS

6. Solve the following Absolute-Value Equation:
Practice:
1) Solve 6x-3 = 15
2) Solve 2x = 8

7. * Plug in answers to check your solutions!
1) Solve 6x-3 = 15
6x-3 = or 6x-3 = -15
6x = or 6x = -12
x = 3 or x = -2
* Plug in answers to check your solutions!

8. Get the abs. value part by itself first!
Answer ::
2) Solve 2x = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.

9. ***Important NOTE*** 3 2x + 9 +12 = 10 - 12 - 12 3 2x + 9 = - 2 3 3
3 2x = - 2
No Solution
2x = - 2 3
What about this absolute value equation? 3x – 6 – 5 = – 7

10. Solving & Graphing Absolute Value Inequalities

11. Solving an Absolute Value Inequality:
Step 1: Rewrite the inequality as a conjunction or a disjunction.
If you have a you are working with a conjunction or an ‘and’ statement.
Remember: “Less thand”
If you have a you are working with a disjunction or an ‘or’ statement.
Remember: “Greator”
Step 2: In the second equation you must negate the right hand side and reverse the direction of the inequality sign.
Solve as a compound inequality.

12. Ex: “and” inequality 4x – 9 ≤ 21 4x – 9 ≥ -21 + 9 + 9 + 9 + 9 4x ≤ 30
Becomes an “and” problem
Positive
Negative
4x – 9 ≤ 21
4x – 9 ≥ -21
4x ≤ 30
4x ≥ -12
x ≤ 7.5
x ≥ -3

13. This is an ‘or’ statement. (Greator).
Ex: “or” inequality
In the 2nd inequality, reverse the inequality sign and negate the right side value.
|2x + 1| > 7
2x + 1 > or x + 1 < - 7 – –
2x > 6
2x < - 8
x < - 4
x > 3 3 -4

14. Solving Absolute Value Inequalities:
Solve | x  4 | < 3 and graph the solution.
Solve | 2x  1 | 3  6 and graph the solution.

15. Answer :: Solve | x  4 | < 3 Reverse inequality symbol.
x  4 IS POSITIVE
x  4 IS NEGATIVE
| x  4 |  3
| x  4 |  3 
x  4  3
x  4  3
Reverse inequality symbol.
x  7
x  1
The solution is all real numbers greater than 1 and less than 7.
This can be written as 1  x  7.

16. Solve | 2x  1 | 3  6 and graph the solution.
Answer ::
Solve | 2x  1 | 3  6 and graph the solution.
| 2x  1 |  3  6
| 2x  1 |  9
2x  1  +9
x  4
2x  8
| 2x  1 | 3  6
2x  1  9
2x  10
x  5
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
Reverse inequality symbol.
The solution is all real numbers greater than or equal to 4 or less than or equal to  5. This can be written as the compound inequality x   5 or x  4.
 6  5  4  3  2 

17. Solve and graph the following Absolute-Value Inequalities:3)
|x -5| < 3

18. Answer :: Solve & graph. 3) Get absolute value by itself first.
Becomes an “or” problem

19. Answer :: |x -5|< 3 x -5< 3 and x -5< 3
This is an ‘and’ statement.
(Less thand).
Rewrite.
In the 2nd inequality, reverse the inequality sign and negate the right side value.
Solve each inequality.
Graph the solution.
|x -5|< 3
x -5< 3 and x -5< 3
x -5< 3 and x -5> -3
x < 8 and x > 2
2 < x < 8 8 2

20. Steps