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Solving Absolute Value Equations and Inequalities Made Easy

This guide provides clear examples and solutions for absolute value equations and inequalities. Learn how to solve problems such as |x| = 5, |x| > 5, and |x| ≤ 5, along with important rules to apply. The content covers both simple cases and more complex scenarios, making it suitable for beginners and advanced learners alike. Find step-by-step solutions and graphical representations to enhance your understanding of absolute value concepts in mathematics.

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Solving Absolute Value Equations and Inequalities Made Easy

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  1. 1.5 “absolute value equations & inequalities”

  2. ex 1: Solve |x| = 5

  3. ex 1: Solve |x| = 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  4. ex 1: Solve |x| = 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x = 5

  5. ex 1: Solve |x| = 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x = 5

  6. ex 1: Solve |x| = 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x = 5 or x = –5

  7. ex 1: Solve |x| = 5 |x| = a rule: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  8. ex 1: Solve |x| = 5 |x| = a rule: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x = a “or” x = –a

  9. ex 2: Solve |x| > 5

  10. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  11. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  12. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  13. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  14. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  15. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  16. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  17. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  18. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  19. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  20. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  21. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  22. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5

  23. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5 or x < –5

  24. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5 or x < –5

  25. ex 2: Solve |x| > 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > 5 or x < –5

  26. ex 2: Solve |x| > 5 |x| > a rule: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  27. ex 2: Solve |x| > 5 |x| > a rule: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > a “or” x < –a

  28. ex 3: Solve |x|  5

  29. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  30. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  31. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  5

  32. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  5

  33. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  5 and x  –5

  34. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  5 and x  –5

  35. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  5 and x  –5

  36. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  –5 and x  5

  37. ex 3: Solve |x|  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  –5 and x  5

  38. ex 3: Solve |x|  5 –5  x  5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x  –5 and x  5

  39. ex 3: Solve |x|  5 |x| < a rule: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

  40. ex 3: Solve |x|  5 |x| < a rule: -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > –a “and” x < a

  41. rules:

  42. rules: |ax + b| = c

  43. rules: |ax + b| = c ax + b = c or ax + b = –c

  44. rules: |ax + b| = c ax + b = c or ax + b = –c |ax + b| > c

  45. rules: |ax + b| = c ax + b = c or ax + b = –c |ax + b| > c ax + b > c or ax + b < –c

  46. rules: |ax + b| = c ax + b = c or ax + b = –c |ax + b| > c ax + b > c or ax + b < –c |ax + b| < c

  47. rules: |ax + b| = c ax + b = c or ax + b = –c |ax + b| > c ax + b > c or ax + b < –c |ax + b| < c ax + b < c and ax + b > –c

  48. ex 6: Solve |x – 4| = 2

  49. ex 6: Solve |x – 4| = 2 x – 4 = 2 or x – 4 = –2

  50. ex 6: Solve |x – 4| = 2 x – 4 = 2 or x – 4 = –2 x = 6 or x = 2

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