1. Absolute Value Inequalities
Algebra

2. Solving an Absolute-Value Inequalities
 8  7  6  5  4  3  2 
 8  7  6  5  4  3  2 

3. Graphing Absolute Value
When an absolute value is greater than the variable you have a disjunction to graph.
When an absolute value is less than the variable you have a conjunction to graph.

4. This can be written as 1  x  7.
Solving an Absolute-Value Inequality
Solve | x  4 | < 3
x  4 IS POSITIVE
x  4 IS NEGATIVE
| x  4 |  3
| x  4 |  3
x  4  3
x  4  3 
x  7
x  1
Reverse inequality symbol.
The solution is all real numbers greater than 1 and less than 7.
This can be written as 1  x  7.

5. Solve | 2x  1 | 3  6 and graph the solution.
Solving an Absolute-Value Inequality
| 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
Solve | 2x  1 | 3  6 and graph the solution.
2x + 1 IS POSITIVE
2x + 1 IS NEGATIVE
| 2x  1 |  3  6
| 2x  1 | 3  6
| 2x  1 |  9
| 2x  1 |  9
2x  1  +9
2x  1  9 
2x  8
2x  10
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.
x  4
Reverse inequality symbol.
x  5
 5 4.
 6  5  4  3  2 

6. Strange Results
True for All Real Numbers, since absolute value is always positive, and therefore greater than any negative.
No Solution Ø. Positive numbers are never less than negative numbers.

8. Absolute Value Inequalities
Algebra