IGCSE Maths / Inequalities
Whole-number solutions to an inequality revision notes
Edexcel International GCSE Maths (4MA1), chapter 8: Inequalities, specification point 2.8b-f. Free lesson.
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To find whole-number solutions, first solve the inequality, then list every integer that fits. Check each end separately, because a strict sign excludes its end while a sign with a line underneath includes it.
How to list solutions
An integer is a whole number, including negative numbers and zero. Once you know the range that satisfies an inequality, list every integer in it. The symbols at the ends tell you whether those end values count. A strict sign, < or >, leaves that end out. A sign with a line underneath, ≤ or ≥, includes that end. Make the list complete, with no excluded end point.
Check each end of a range
For −2 < x ≤ 3, check the left and right ends separately. The left sign is strict, so x cannot equal −2. The right sign includes equality, so x can equal 3. The integers between the ends are −1, 0, 1, 2 and 3. There are five solutions, and zero counts.
Solve before listing
When an inequality needs solving, find its range first. Then combine it with any other condition on x. Here, x ≤ 3 combines with x > 0, so zero is excluded and the solutions are 1, 2 and 3.
Whole-number solutions include zero and negatives
Whole-number solutions here means integer solutions, not just positive counting numbers. For example, from −1 to 3 inclusive, the integers are −1, 0, 1, 2 and 3. Fractions and decimals between them are not included. Do not leave out zero or negative integers when they fit the inequality. For −4 ≤ x < 2, include −4 because its sign has a line underneath, but exclude 2 because its sign is strict. The solutions are −4, −3, −2, −1, 0, 1.