Solving Absolute Value Inequalities
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  • 1. Solving Absolute Value Equations
    Goals:
    Solve equations involving absolute values
    Rewrite absolute value inequalities in compound form and solve
  • 2. Definition
    The absolute value of a number, x, is the distance the number is from 0.
    Since distance is nonnegative, the absolute value of a number is always nonnegative.
    x
    , if x is positive
  • 3. Definition
    The absolute value of a number, x, is the distance the number is from 0.
    Since distance is nonnegative, the absolute value of a number is always nonnegative.
    x
    , if x is positive
    0
    , if x is 0
  • 4. Definition
    The absolute value of a number, x, is the distance the number is from 0.
    Since distance is nonnegative, the absolute value of a number is always nonnegative.
    x
    , if x is positive
    0
    , if x is 0
    -x
    , if x is negative
  • 5. If you take the negative of an absolute value,
    Then your answer becomes negative.
    !!!CAUTION!!!
  • 6. , then the x can be equal
    If the
    to either 6 or -6.
    Therefore, when taking the absolute value of an equation, there are TWO possible solutions:
    OR
    Absolute Value Equations
  • 7. Now solve both equations separately:
    Write your solutions together:
    x = -7 or 3
    Absolute Value Equations
  • 8. EXAMPLES:
    1)
    2)
    3)
    4)
  • 9. EXAMPLES:
    1)
    2)
    3)
    4)
    x = -2 or 8
  • 10. EXAMPLES:
    1)
    2)
    3)
    4)
    x = -2 or 8
    x = -6 or 1
  • 11. EXAMPLES:
    1)
    2)
    3)
    4)
    x = -2 or 8
    x = -6 or 1
    x = -10 or -6
  • 12. EXAMPLES:
    1)
    2)
    3)
    4)
    x = -2 or 8
    x = -6 or 1
    x = -10 or -6
    -11
    2
    x = -4 or
  • 13. The way to solve multi-step absolute value equations is like solving a multi-step linear equation.
    Instead of solving by isolating the variable by undoing
    1st: addition/subtraction
    2nd: multiplication/division
    You isolate the absolute value by undoing
    1st: addition/subtraction outside the abs. value
    2nd: multiplication/division outside the abs. value
    Multi-Step Absolute Value Equations
  • 14. Walk-Through: Solve the following
    -7
    3
    x = 3 or
  • 15. x = -12 or -2
    Walk-Through: Solve the following
  • 16. Solve the following absolute value equations:
    1)
    2)
    3)
    Try the Following On Your Own:
  • 17. Solve the following:
    1)
    x = or 4
    2)
    3)
    -2
    3
    Solutions:
  • 18. Solve the following:
    1)
    x = or 4
    2)
    x = or
    3)
    -2
    3
    -9
    2
    19
    2
    Solutions:
  • 19. Solve the following:
    1)
    x = or 4
    2)
    x = -1 or 6
    3)
    x = -6 or 0
    -2
    3
    Solutions: