Pres Absolute Value Inequalities (Section 1.8)

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  • 1. Absolute Value
  • 2. Absolute Value
  • 3. Rules and Properties
    • Absolute-Value Equations
    • If a > 0 and | x | = a , then x = a or x = – a .
    1.8 Solving Absolute-Value Equations and Inequalities Absolute-Value Inequalities If a > 0 and | x | < a , then x > – a and x < a . If a > 0 and | x | > a , then x < – a or x > a . Similar statements are true for | x |  a and | x |  a.
  • 4. Absolute Value Rules
    • If you start with an “=“ sign then you will have an “or” statement.
    • If you start with a “<“ or a “ ≤” then you will have an “and” statement.
    • If you start with a “>” or a “≥” then you will have an “or” statement.
  • 5. Example 1
    • Solve and graph absolute-value equations.
    | x + 1| = 7 1.8 Solving Absolute-Value Equations and Inequalities
  • 6. Example 1
    • Solve and graph absolute-value equations.
    | x + 1| = 7 1.8 Solving Absolute-Value Equations and Inequalities x + 1 = 7 or x + 1 = –7 x = 6 or x = –8
  • 7. Example 2
    • Solve and graph absolute-value equations.
    4| x | = 8 1.8 Solving Absolute-Value Equations and Inequalities
  • 8. Example 2
    • Solve and graph absolute-value equations.
    4| x | = 8 1.8 Solving Absolute-Value Equations and Inequalities 4 x = 8 or 4 x = –8 x = 2 or x = –2
  • 9. Example 3 |x - 4| = x + 1
  • 10. Example 3 |x - 4| = x + 1 x = x + 5 or x – 4 = -x - 1 0 = 5 or 2x – 4 = -1 x - 4 = x + 1 x - 4 = -(x + 1) No Solution or 2x = 3 No Solution or x = 3/2 3/2
  • 11. Example 4
    • Solve and graph absolute-value inequalities.
    | x + 52|  76 1.8 Solving Absolute-Value Equations and Inequalities
  • 12. Example 4
    • Solve and graph absolute-value inequalities.
    | x + 52|  76 x + 52  76 or x + 52  - 76 x  24 or x  – 128 1.8 Solving Absolute-Value Equations and Inequalities
  • 13. Example 5
    • Solve and graph absolute-value inequalities.
    | x + 52|  76 1.8 Solving Absolute-Value Equations and Inequalities
  • 14. Example 5
    • Solve and graph absolute-value inequalities.
    | x + 52|  76 x + 52  76 and x + 52  -76 x  24 and x  –128 1.8 Solving Absolute-Value Equations and Inequalities