Solve Equations with Absolute Value




This is a tutorial on solving equations with absolute value. Detailed solutions and explanations are included.

Example 1: Solve the equation

|x + 6 | = 7


Solution to Example 1:

  • If |x + 6 | = 7, then
    a)x + 6 = 7
    or
    b)x + 6 = -7

  • Solve equation a)
    x + 6 = 7
    x = 1

  • Solve equation b)
    x + 6 = -7
    x = -13

Check solutions:

  • solution x = 1
    Left Side of Equation for x = 1.
    |1 + 6 |
    = | 7 |
    = 7
    Right Side of Equation for x = 1.
    7
  • x = -13
    Left Side of Equation for x = 1.
    |-13 + 6 |
    = | -7 |
    = 7
    Right Side of Equation for x = 1.
    7

The solutions to the given equation are x = 1 and x = -13

Matched Exercise 1: Solve the equation

|-x - 8 | = 10

Answers

Example 2: Solve the equation

-2 |x / 2 + 3 | - 4 = -10

Solution to Example 2:

  • Given
    -2 |x / 2 + 3 | - 4 = -10

  • We first write the equation in the form | A | = B. Add 4 to both sides and group like terms
    -2|x / 2 + 3 | = -6

  • Divide both sides by -2
    |x / 2 + 3 | = 3

  • We now proceed as in example 1 above, the equation
    |x / 2 + 3 | = 3 gives two equations.

    a)x / 2 + 3 = 3
    or
    b)x / 2 + 3 = -3

  • Solve equation a)
    x / 2 + 3 = 3

  • to obtain
    x = 0

  • Solve equation b)
    x / 2 + 3 = -3

  • to obtain
    x = -12

Check solutions:

  • x = 0
    Left Side of Equation for x = 0.
    -2 |x / 2 + 3 | - 4
    = -2| 3 | - 4
    = -10
    Right Side of Equation for x = 1.
    -10
  • x = -12
    Left Side of Equation for x = -12.
    -2 |x / 2 + 3 | - 4
    = -2 |-12 / 2 + 3 | - 4
    = -2 |-6 + 3 | - 4
    = -2(3) - 4
    = -10
    Right Side of Equation for x = -12.
    -10

The solutions to the given equation are x = 0 and x = -12

Matched Exercise 2: Solve the equation

4 |x + 2| - 30 = -10

Answers


Example 3: Solve the equation

|2 x - 2 | = x + 1

Solution to Example 3:

  • If 2 x - 2 >= 0 which is equivalent to x >= 1, then |2 x - 2 | = 2 x - 2 and the given equation becomes
    2 x - 2 = x + 1

  • Add 2 - x to both sides
    x = 3

  • Since x = 3 satisfies the condition x >= 1, it is a solution.
  • If 2x - 2 < 0 which is equivalent to x < 1, then |2 x - 2 | = -(2 x - 2) and the given equation becomes
    -(2 x - 2) = x + 1

  • Solve for x to obtain
    x = 1 / 3

  • Since x = 1 / 3 satisfies the condition x < 1, it is a solution.

Check solutions

  • x = 3
    Left Side of Equation for x = 3.
    |2 x - 2 |
    = |2*3 - 2 |
    = 4
    Right Side of Equation for x = 3.
    x + 1
    = 3 + 1
    = 4
  • x = 1/3
    Left Side of Equation for x = 1 / 3.
    |2 x - 2 |
    = |2*(1/3) - 2 |
    = 4 / 3
    Right Side of Equation for x = 1 / 3.
    x + 1
    = 4 / 3

The solutions to the given equation are x = 3 and x = 1 / 3

Matched Exercise 3:Solve the equation

- 4|x + 2 | = x - 8

Answers


Example 4: Solve the equation

|x2 - 4| = x + 2

Solution to Example 3:

  • If x2 - 4 >= 0 ,or x2 >= 4, then | x2 - 4 | = x2 - 4 and the given equation becomes
    x2 - 4 = x + 2

  • Add - (x + 2) to both sides
    x2 - 4 -( x + 2) = 0

  • Factor the left term
    (x - 2)(x + 2) -( x + 2) = 0

    (x + 2)(x - 2 -1) = 0

    (x + 2)(x - 3) = 0

  • Using the factor theorem, we can write two simpler equations
    x + 2 = 0
    or
    x - 3 = 0

  • Solve the above equations for x to find two values of x that make the left side of the equation equal to zero.
    x = -2 and x = 3.

  • Both values satisfy the condition x2 >= 4 and are solutions to the given equation.
    x = -2 and x = 3.

  • If x2 - 4 < 0 ,or x2 < 4, then | x2 - 4 | = -(x2 - 4) and the given equation becomes.
    -(x2 - 4) = x + 2

    -(x2 - 4) - ( x + 2) = 0

  • Factor the left term.
    -(x - 2)(x + 2) - ( x + 2) = 0

    (x - 2)(x + 2) + ( x + 2) = 0

    (x - 2)(x + 2) + ( x + 2) = 0

    (x + 2)(x - 2 + 1) = 0

    (x + 2)(x - 1) = 0

  • Two values make the left side of the above equation equal to zero
    x = -2 and x = 1.

  • Only x = 1 satisfies the condition x2 < 4

Check solutions:

  • x = -2
    Right Side of Equation = | x2 - 4 |
    = | (-2)2 - 4 | = 0
    Left Side of Equation = x + 2 = -2 + 2 = 0

  • x = 3 Left Side of Equation = | x2 - 4 |
    = | 32 - 4 |
    = | 5 |
    = 5 Right Side of Equation = x + 2 = 3 + 2 = 5
  • x = 1
    Left Side of Equation = | x2 - 4 |
    = | 12 - 4 | = | - 3 | = 3 Right Side of Equation = x + 2 = 1 + 2 = 3

Conclusion

The solutions to the given equation are x = -2, x = 1 and x = 3.

Matched Exercise 4: Solve the equation

|x2 - 16 | = x - 4

Answers





Exercises.(see answers below)

Solve the following absolute value equations

a) | x - 4 | = 9

b) | x 2 + 4 | = 5

c) | x 2 - 9 | = x + 3

d) | x + 1 | = x - 3

e) | -x | = 2

Answers to Above Exercises.

a) -5 , 13

b) -1 , 1

c) -3 , 2 , 4

d) no real solutions

e) -2 , 2

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Updated: 2 April 2013

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