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Absolute Value

Solving Equations Containing Absolute Value

Problems

Problems

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.

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