Solving Equations Containing Absolute Value
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:
- Perform inverse operations until the absolute value stands by itself on one side of
the equation--the equation should be of the form|expression| = c.
If c is negative, the equation has no solution.
- Separate into two equations: expression = c or expression = -c
Note that "or" implies a union of the two equations.
- Solve both equations to yield the two solutions: x = a and x = b
- Check the solutions in the original equation.
Example 1: Solve for x: | 2x - 1| + 3 = 6.
- Perform inverse operations: | 2x - 1| = 3
- Separate: 2x - 1 = 3 or 2x - 1 = - 3
- Solve:
2x - 1 = 3
2x = 4
x = 2
or 2x - 1 = - 3
2x = - 2
x = - 1
x = 2 or x = - 1
- Check: | 2(2) - 1| + 3 = 6 ? Yes. | 2(- 1) - 1| + 3 = 6 ? Yes.
Thus,
x = - 1, 2.
Example 2: Solve for x: 
= 7.
- Perform inverse operations: | x - 1| = 21
- Separate: x - 1 = 21 or x - 1 = - 21
- Solve:
x - 1 = 21
x = 22
or x - 1 = - 21
x = - 20
x = 22 or x = - 20
- Check:
= 7 ? Yes. 
= 7 ? Yes.
Thus,
x = - 20, 22
Example 3: Solve for x: | 2x - 1| + 7 = 5.
- Perform inverse operations: | 2x - 1| = - 2
The absolute value of a quantity cannot be negative, so the equation has no
solution.
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