Double Inequalities for IGCSE Maths

What You Need to Know

Solving inequalities with two bounds and listing integer solutions. This subtopic is part of Inequalities in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding double inequalities is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Double Inequalities

A double inequality has the form a < f(x) ≤ b, meaning f(x) satisfies both inequalities simultaneously. The strategy is to perform the same operation to all three parts of the inequality to isolate x in the middle. Results are written as a single combined inequality and represented on a number line with the appropriate circle types at each endpoint. Tested on both Core and Extended papers.

Step-by-Step Method

1. 1

   Write the double inequality

   Identify all three parts: left bound, middle expression with x, right bound.

2. 2

   Apply operations to all three parts simultaneously

   Whatever you do to the middle, do to the left and right as well: add, subtract, multiply, or divide all three.

3. 3

   Reverse both inequality signs if dividing/multiplying by negative

   Both < become >, and vice versa, when you divide all three parts by a negative number.

4. 4

   Write the final combined inequality

   Express as a < x < b (or similar with ≤ and ≥).

5. 5

   Find integer solutions if needed

   List all whole numbers satisfying the inequality.

Exam Tips for Double Inequalities

• Perform the same operation to ALL THREE parts — a common error is operating on only two of them.
• When dividing by a negative, BOTH inequality signs reverse simultaneously.
• List integer solutions carefully — check the boundary values: if x < 4, then 4 is NOT included.
• Represent on a number line with correct open/closed circles at each bound.

Practice Questions

Frequently Asked Questions