Absolute Value Equation
See how to solve an absolute value equation (one variable).
6 examples and their solutions.
Absolute Value
Definition
|(+)| = (+)
|0| = 0
|(-)| = (+)
The absolute value means|0| = 0
|(-)| = (+)
how big the number is.
'| |' is the absolute value sign.
So the absolute value
ignores the (-) sign.
|(-)| = (+)
Example
|5|
Solution |5| = 5
Example
|-2|
Solution |-2| = 2
Absolute Value Equation (One Variable)
Example
|x - 1| = 2
Solution |x - 1| = 2
1) x - 1 ≥ 0
x - 1 = 2 - [1]
x = 3
2) x - 1 < 0
-(x - 1) = 2 - [2]
-x + 1 = 2
-x = 1
x = -1
x = 3, -1
1) x - 1 ≥ 0
x - 1 = 2 - [1]
x = 3
2) x - 1 < 0
-(x - 1) = 2 - [2]
-x + 1 = 2
-x = 1
x = -1
x = 3, -1
[1]
x - 1 ≥ 0
|x - 1| = 2
→ x - 1 = 2
|x - 1| = 2
→ x - 1 = 2
[2]
x - 1 < 0
|x - 1| = 2
→ -(x - 1) = 2
|x - 1| = 2
→ -(x - 1) = 2
Close
Example
|2x - 3| + 1 = 0
Solution |2x - 3| + 1 = 0
|2x - 3| = -1 - [1]
No solution
|2x - 3| = -1 - [1]
No solution
[1]
The left side, |2x - 3|,
is an absolute value sign.
So the left side is either 0 or (+).
But the right side, -1, is (-).
So there's no solution.
is an absolute value sign.
So the left side is either 0 or (+).
But the right side, -1, is (-).
So there's no solution.
Close
Absolute Value Inequality (One Variable)
Formula
|x| < a
→ -a < x < a
|x| > a
→ 1) x < -a
2) x > a
→ -a < x < a
|x| > a
→ 1) x < -a
2) x > a
Example
|x - 2| < 5
Solution |x - 2| < 5
-5 < x - 2 < 5
-3 < x < 7 - [1]
-5 < x - 2 < 5
-3 < x < 7 - [1]
[1]
+2 to each side.
Close
Example
|2x + 1| ≥ 9
Solution |2x + 1| ≥ 9
1) 2x + 1 ≤ -9
2x ≤ -10
x ≤ -5
2) 2x + 1 ≥ 9
2x ≥ 8
x ≥ 4
x ≤ -5 or x ≥ 4
1) 2x + 1 ≤ -9
2x ≤ -10
x ≤ -5
2) 2x + 1 ≥ 9
2x ≥ 8
x ≥ 4
x ≤ -5 or x ≥ 4
Close