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2.7PureStretch

Solving Modulus Problems

Solving equations and inequalities with a modulus needs care, because can equal a value in two ways. A sketch plus the two-case approach finds every solution — and helps you discard false ones.

30 min Video by Zeeshan Zamurred Functions and Graphs
Edexcel A level Maths: 2.7 Solving Modulus ProblemsWatch the full walkthrough before the notes below.
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What you'll be able to do

  • Solve equations of the form |f(x)| = g(x)
  • Use the two-case method (±)
  • Use a sketch to find and check solutions
  • Solve modulus inequalities
1

The two cases

Because means , solve both cases. Always each solution in the original equation, since the modulus can introduce false answers.

Two cases — then verify.
1Case 1: .
2Case 2: .
Answer or
2

Use a sketch

For , sketch both and ; the solutions are the intersection points. A sketch shows how many solutions to expect and rules out impossible cases (e.g. a modulus can never equal a negative).

Tip — |f(x)| = (a negative number) has NO solutions — the modulus is never negative.

3

Modulus inequalities

For inequalities like , the sketch (or the rule ) gives the solution interval. Reading the regions from a clear sketch is the most reliable method.

1 means .
Answer

Formula recap

Two-case method.
Modulus inequality.
No solution if RHS < 0.

Common mistakes to avoid

Only solving the positive case.
Solve both f(x) = a and f(x) = −a, then check.
Keeping a solution that fails the original equation.
Always verify — the ± method can produce false solutions.

Key takeaways

  • |f(x)| = a gives two cases: f(x) = a or f(x) = −a (then verify).
  • Sketch y = |f(x)| and y = g(x); solutions are the intersections.
  • |x| < a means −a < x < a; a modulus is never negative.

Test yourself

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