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MathidiaMathematics Academy
Advanced algebra and functions
Lesson 4 of 75
Challenge

Modulus equations and inequalities

Objective. Solve algebraically and interpret graphically.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Modulus equations and inequalities involve |expression|, the non-negative distance of a value from zero, solved either by squaring both sides or by splitting into cases based on sign. Example: |2x−1|=5 gives 2x−1=5 or 2x−1=−5, so x=3 or x=−2. For inequalities like |x−2|<3, this means −3<x−2<3, giving −1<x<5; sketching y=|f(x)| alongside the other side of the equation helps confirm all solutions graphically.

Worked example

Solve the modulus equation |3x+2|=8.

3 … 2
Illustrated card showing 3 … 2
  1. 1Split into two cases: 3x+2=8 or 3x+2=−8
  2. 2Solve the first case: 3x=6, so x=2
  3. 3Solve the second case: 3x=−10, so x=−10/3
  4. 4Check both values satisfy the original modulus equation

Answer: x=2 or x=−10/3

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Common mistake. A frequent mistake is solving |x|=k by only taking the positive case, missing x=−k; always write both x=k and x=−k before proceeding further.

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