Download Absolute Value Inequalities

Absolute
Value
Inequalities
Algebra
Solving an Absolute-Value Inequalities
x 6
8 7 6 5 4 3 2 1
0
1
2
3
4
5
6
7
8
x 6
8 7 6 5 4 3 2 1
0
1
2
3
4
5
6
7
8
Graphing Absolute Value
• When an absolute value is greater than
the variable you have a disjunction to
graph.
x 4
• When an absolute value is less than
the variable you have a conjunction to
graph.
x 4
Solving an Absolute-Value Inequality
Solve | x  4 | < 3
x  4 IS POSITIVE
|x4|3
x  4  3
x7
x  4 IS NEGATIVE
|x4|3
x  4  3
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.
Solving an Absolute-Value Inequality
Solve
 1POSITIVE
| 3  6 and graph
2x +| 2x
1 IS
2x +the1solution.
IS NEGATIVE
| 2x  1 |  3  6
2x + 1 IS POSITIVE
| 2x|2x
1|
 31 |6 9
2x 1 |1 
| 2x
9 +9
| 2x  1 | 3  6
2x + 1 IS NEGATIVE
| 2x|2x
1|
31 |6  9
2x

1

9
| 2x  1 |  9
2x

10

2x  1 9
x  5
2x  8
x4
2x  1  +9
2x  10
2x  8
The solution is all real numbers greater than or equal
x4
x  5
to 4 or less than or equal Reverse
to  5. This can be written as
the compound inequality inequality
x   5 or x  4.
symbol.
6 5 4 3 2 1
0
1
2
3
4
5
6
Strange Results
2(3x  8)  7  5
(2[3x  (8  4)]  12)3  2
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.
Absolute
Value
Inequalities
Algebra