The equation
| x| = 4
means
x = 4
or
x = - 4
.
The equation
| x - 12| = 4
means
x - 12 = 4
or
x - 12 = - 4
. Thus,
x = 16
or
x = 8
.
Check: | 16 - 12| = 4 ? Yes. | 8 - 12| = 4 ? Yes.The equation | x + 2| - 1 = 8 can be solved in a similar manner:
| x + 2| - 1 + 1 = 8 + 1
| x + 2| = 9
x + 2 = 9 or x + 2 = - 9
x + 2 - 2 = 9 - 2 or x + 2 - 2 = - 9 - 2
x = 7 or x = - 11
In general, to solve an equation with an absolute value:
Example 1: Solve for
x
:
| 2x - 1| + 3 = 6
.
2x - 1 = 3x = 2 or x = - 1
2x = 4
x = 2
or 2x - 1 = - 3
2x = - 2
x = - 1
Example 2: Solve for
x
:
= 7
.
x - 1 = 21x = 22 or x = - 20
x = 22
or x - 1 = - 21
x = - 20
Example 3: Solve for
x
:
| 2x - 1| + 7 = 5
.
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