Modulus equations and inequalities

Two cases

|f(x)| = g(x) means f(x) = g(x) or -f(x) = g(x). Solve both, then check each answer in the original equation: a modulus cannot equal a negative number, so reject any x where g(x) < 0.

Use a sketch

Sketch y = |f(x)| and y = g(x). The number of intersection points is the number of solutions. The specification expects you to use graphs like this to solve modulus equations and inequalities, such as |2x - 1| = x + 5.

Inequalities

• |x - a| < b means -b < x - a < b.
• |x - a| > b means x - a < -b or x - a > b.

For harder ones, find where the graphs meet, then read from the sketch where one graph is under or over the other.

Worked example

Solve |2x - 1| = x + 5.

1. Case 1: 2x - 1 = x + 5 gives x = 6.
2. Case 2: -(2x - 1) = x + 5 gives -3x = 4, so x = -4/3.
3. Both give x + 5 > 0, so both are valid.

Answer: x = 6 or x = -4/3

Check yourself

1. Solve |x - 3| = 5.
   Show answer

   x = 8 or x = -2. x - 3 = 5 or x - 3 = -5.

2. Solve |3x + 2| = 2x + 8.
   Show answer

   x = 6 or x = -2. 3x + 2 = 2x + 8 gives x = 6. -(3x + 2) = 2x + 8 gives -5x = 10, so x = -2. Both make 2x + 8 positive.

3. How many solutions does |2x - 5| = x - 4 have?
   Show answer

   0. The cases give x = 1 and x = 3, but at both x - 4 is negative, so neither works. A sketch shows the line stays under the V.

4. Solve |x - 4| < 3.
   • x < 1 or x > 7
   • -7 < x < 7
   • 1 < x < 7
   • x < 7
   Show answer

   1 < x < 7. -3 < x - 4 < 3, so 1 < x < 7.

5. Solve |2x + 1| ≥ 5.
   • x ≤ -3 or x ≥ 2
   • -3 ≤ x ≤ 2
   • x ≥ 2
   • x ≤ -2 or x ≥ 3
   Show answer

   x ≤ -3 or x ≥ 2. 2x + 1 ≥ 5 gives x ≥ 2; 2x + 1 ≤ -5 gives x ≤ -3.

6. Solve |x| > 2x - 3.
   • x > 3
   • x < 1 or x > 3
   • x < 3
   • 1 < x < 3
   Show answer

   x < 3. For x ≥ 0: x > 2x - 3 gives x < 3. For x < 0: -x > 2x - 3 gives x < 1, which every negative x satisfies. Together: x < 3.

7. Solve |x² - 5| = 4.
   Show answer

   x = ±3 or x = ±1. x² - 5 = 4 gives x² = 9; x² - 5 = -4 gives x² = 1.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.