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Solving Equations Containing Absolute Value

Solving Equations Containing Absolute Value

Solving Equations Containing Absolute Value

Solving Equations Containing Absolute Value

Solving Equations Containing Absolute Value

Solving Equations Containing Absolute Value

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.