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MathidiaMathematics Academy
Functions and graph transformations
Lesson 10 of 75
Review

Modulus functions

Objective. Sketch and solve equations involving modulus graphs.

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

Learn: the key idea

The modulus function |x| returns the non-negative value of x, so |f(x)| reflects any part of y=f(x) below the x-axis upward, while y=f(|x|) reflects the right-hand side of the graph onto the left. To solve equations like |2x−1|=5, split into two cases: 2x−1=5 or 2x−1=−5, giving x=3 or x=−2. When sketching |f(x)|, keep the positive sections unchanged and mirror only the negative sections. Always check each solution against the original equation since squaring or splitting can introduce extraneous values.

Worked example

Solve the modulus equation |3x+2| = 7 for all values of x.

3 … 2
Illustrated card showing 3 … 2
  1. 1Split into two cases: 3x+2=7 and 3x+2=−7
  2. 2Solve the first case: 3x=5, so x=5/3
  3. 3Solve the second case: 3x=−9, so x=−3
  4. 4State both solutions: x=5/3 and x=−3

Answer: x = 5/3 or x = −3

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Common mistake. Students solve |x−2|=3 by writing only x−2=3, missing the second case x−2=−3; both branches must be considered, giving solutions x=5 and x=−1, not just one.

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