Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions

Solving linear and quadratic inequalities is a fundamental skill in A-Level Mathematics, extending your algebraic manipulation to handle ranges of values rather than exact solutions. This topic involves finding the set of values for a variable that satisfy an inequality, such as 2x + 3 > 7 or x² - 4x < 0. You'll learn to solve these algebraically and graphically, interpreting the solutions on number lines or coordinate axes. Inequalities often appear in real-world contexts like optimisation problems, and they are essential for later topics such as domain and range of functions, calculus, and modelling.

For linear inequalities, the process is similar to solving equations, but you must be careful when multiplying or dividing by a negative number, which reverses the inequality sign. Quadratic inequalities require factorising or using the quadratic formula to find critical values, then testing intervals to determine where the inequality holds. Graphically, you can sketch the quadratic curve and identify where it lies above or below the x-axis. This visual approach reinforces the algebraic method and helps avoid sign errors.

This topic builds on GCSE algebra and is a prerequisite for more advanced work in A-Level Mathematics, including solving systems of inequalities, linear programming, and calculus applications like finding intervals of increase or decrease. Mastery of inequalities also supports problem-solving in mechanics and statistics. By the end of this topic, you should be able to solve any linear or quadratic inequality, represent the solution set on a number line, and interpret the graphical meaning.