Inequality definition
An inequality is a mathematical statement that compares two values or expressions when they are not necessarily equal. Instead of using only an equals sign, an inequality uses comparison symbols such as less than, greater than, less than or equal to, and greater than or equal to. For example, x is greater than 5 means that x can be any value bigger than 5, not just one exact number.
The main inequality symbols are important to recognise. The symbol for less than means the value on the left is smaller. The symbol for greater than means the value on the left is larger. The symbols less than or equal to and greater than or equal to include the boundary value as well. For example, x is less than or equal to 4 includes 4, while x is less than 4 does not include 4.
Inequalities are used in algebra, graphs, number lines, functions, probability, statistics, optimisation, real-life modelling and constraints. At GCSE level, students solve linear inequalities, show solutions on number lines, use integer solutions and compare algebraic expressions. At A-Level, inequalities appear in functions, modulus questions, regions on graphs, calculus applications and proof-style reasoning.
A simple inequality might be x + 3 is less than 10. To solve it, subtract 3 from both sides to get x is less than 7. The solution is not one number. It is a set of values: any number below 7 will work. This is one major difference between many equations and inequalities. Equations often have exact solutions, while inequalities often describe a range of possible solutions.
Number lines are often used to show inequality solutions. An open circle is used when the boundary value is not included, such as x is greater than 2. A closed circle is used when the boundary value is included, such as x is greater than or equal to 2. An arrow then shows the direction of all possible values. This visual method helps students see the solution set clearly.
When solving inequalities, most steps work like solving equations. You can add, subtract, multiply or divide both sides, as long as the comparison remains true. However, one special rule is very important: when you multiply or divide both sides by a negative number, the inequality sign reverses. For example, if -2x is less than 8, dividing by -2 gives x is greater than -4. Forgetting to reverse the sign is a common exam mistake.
Inequalities are useful in real-life situations because many problems involve limits rather than exact values. A lift may carry at most 500 kilograms, a ticket may cost less than 20 pounds, or a student may need at least 60 percent to pass. These situations naturally use inequality language because they describe boundaries, restrictions and possible ranges.
Related ideas include equations, variables, number lines, intervals, graphs, simultaneous inequalities, regions, linear programming and functions. A clear definition of inequality helps students understand that maths is not always about finding one exact answer. Sometimes it is about describing all values that satisfy a condition. In exams, always check whether the boundary value is included and whether the inequality sign needs to reverse.