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Inequalities with a modulus Cambridge IGCSE Additional Mathematics revision

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In plain words

|x| < 4 says "x is less than 4 away from zero": between −4 and 4. |x| > 4 says "x is more than 4 away from zero": below −4 or above 4. Every modulus inequality is one of these two pictures.

5 things to know

  1. |ax + b| < c means −c < ax + b < c. The solution is a single stretch.
  2. |ax + b| > c means ax + b < −c or ax + b > c. The solution is two separate stretches.
  3. The same holds with ≤ and ≥.
  4. When both sides are moduli, as in |ax + b| ≤ |cx + d|, square both sides. Both are non-negative, so the inequality is kept, and a quadratic inequality is left.
  5. A graph of both sides shows the answer too: find where one graph is below the other.

Worked example

Solve |2x − 1| < 5.

  1. −5 < 2x − 1 < 5.
  2. Add 1 throughout: −4 < 2x < 6.
  3. Divide by 2: −2 < x < 3.

Worked example

Solve |x − 1| ≤ |2x + 3|.

  1. Square both sides: x² − 2x + 1 ≤ 4x² + 12x + 9.
  2. Rearrange: 0 ≤ 3x² + 14x + 8, which is (3x + 2)(x + 4) ≥ 0.
  3. The critical values are −4 and −2/3. The parabola is above the axis outside them: x ≤ −4 or x ≥ −2/3.

Tips and tricks

  • "Less than" gives one region between two values. "Greater than" gives two regions. Getting this the wrong way round is the usual error.
  • Squaring is only safe when both sides are certainly not negative, as two moduli are.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Inequalities with a modulus: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What is the solution of |x| < 4?
    • x < 4
    • −4 < x < 4 (the answer)
    • x < −4 or x > 4
    • x > −4

    x is less than 4 away from zero.

  2. What is the solution of |x| > 2?
    • −2 < x < 2
    • x > 2
    • x < −2 or x > 2 (the answer)
    • x > −2

    x is more than 2 away from zero, on either side.

  3. What is the solution of |x − 3| ≤ 1?
    • 2 ≤ x ≤ 4 (the answer)
    • x ≤ 2 or x ≥ 4
    • −4 ≤ x ≤ −2
    • x ≤ 4

    −1 ≤ x − 3 ≤ 1.

  4. What is the solution of |2x| ≥ 6?
    • −3 ≤ x ≤ 3
    • x ≤ −3 or x ≥ 3 (the answer)
    • x ≥ 3
    • x ≥ 12

    2x ≤ −6 or 2x ≥ 6.

  5. What is the solution of |x + 1| < 3?
    • −2 < x < 4
    • −4 < x < 2 (the answer)
    • x < 2
    • x < −4 or x > 2

    −3 < x + 1 < 3.

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