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:

1. 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.
2. Separate into two equations: expression = c or expression = -c
   Note that "or" implies a union of the two equations.
3. Solve both equations to yield the two solutions: x = a and x = b
4. Check the solutions in the original equation.

Example 1: Solve for x : | 2x - 1| + 3 = 6 .

1. Perform inverse operations: | 2x - 1| = 3
2. Separate: 2x - 1 = 3 or 2x - 1 = - 3
3. Solve:
   

   2x - 1 = 3
   2x = 4
   x = 2
   or 2x - 1 = - 3
   2x = - 2
   x = - 1
   

   x = 2 or x = - 1
4. Check: | 2(2) - 1| + 3 = 6 ? Yes. | 2(- 1) - 1| + 3 = 6 ? Yes.

Thus, x = - 1, 2.

Example 2: Solve for x : = 7 .

1. Perform inverse operations: | x - 1| = 21
2. Separate: x - 1 = 21 or x - 1 = - 21
3. Solve:
   

   x - 1 = 21
   x = 22
   or x - 1 = - 21
   x = - 20
   

   x = 22 or x = - 20
4. Check: = 7 ? Yes. = 7 ? Yes.
   

Thus, x = - 20, 22

Example 3: Solve for x : | 2x - 1| + 7 = 5 .

1. Perform inverse operations: | 2x - 1| = - 2
   The absolute value of a quantity cannot be negative, so the equation has no solution.