Inequalities (Maths) Questions with Solutions

An inequality compares two expressions using the signs <, >, less than or equal to, or greater than or equal to. Unlike an equation, which usually has one or two solutions, an inequality has a whole range of solutions, written on a number line or in interval notation. This topic covers linear inequalities in one and two variables, systems of inequalities, quadratic and rational inequalities, and inequalities involving the modulus, such as |x - 2| < 5. The most important rule is that multiplying or dividing both sides by a negative number reverses the inequality sign. For quadratic and rational inequalities, factorise first, mark the roots on a number line, and test the sign of the expression in each interval. Modulus inequalities convert to compound ones: |x| < a means -a < x < a, while |x| > a means x < -a or x > a. Standard results such as the AM-GM inequality and the triangle inequality also solve many questions on least and greatest values. Practise the inequalities questions below. Each solution shows the number line and the final interval step by step.