Solving Linear Inequalities

Understanding solution ranges in algebra

A linear inequality compares two expressions and shows that one is greater than, less than, greater than or equal to, or less than or equal to the other. Students meet this topic in KS3 and then use it in GCSE and IGCSE algebra. It builds on solving equations, but the answer is usually a range of values rather than one number. Secure work with algebra, equations and negative numbers is helpful before beginning.

Key knowledge and rules

Solve an inequality by using inverse operations on both sides, just as with a linear equation. The essential extra rule is that the inequality sign reverses when both sides are multiplied or divided by a negative number. For example, -2x > 8 becomes x < -4. Review linear equations and the glossary definition of an inequality

Step-by-step method

First simplify both sides if necessary. Second move variable terms to one side and constants to the other. Third divide by the coefficient of the variable. Fourth reverse the sign only if that division is by a negative number. Finally, represent the answer on a number line when requested. Related skills include expanding brackets and factorisation

Worked examples

Example 1: Solve x + 5 < 12. Subtract 5 from both sides to obtain x < 7. Example 2: Solve 3x - 4 ≥ 11. Add 4 to obtain 3x ≥ 15, then divide by 3, giving x ≥ 5. Example 3: Solve -4x ≤ 20. Divide by -4 and reverse the sign, giving x ≥ -5. These methods connect with wider GCSE algebra help

Common mistakes

The most common error is forgetting to reverse the sign after dividing by a negative number. Students may also use a closed circle for a strict inequality or an open circle when the endpoint is included. Checking a value from the proposed range by substitution helps confirm the solution. See the glossary entry for substitution

Practice exercises

Exercise 1: Solve x - 6 > 9. Answer: x > 15. Exercise 2: Solve 5x + 2 ≤ 27. Answer: 5x ≤ 25, so x ≤ 5. Exercise 3: Solve -3x + 4 > 19. Answer: -3x > 15, so after dividing by -3 and reversing the sign, x < -5.

Connections and revision

Inequalities connect naturally with straight-line graphs, simultaneous equations and the meaning of a variable. Students can also strengthen foundations through the main Algebra Lessons

Benefits of one-to-one maths tutoring

One-to-one lessons with maths tutor Ryan Harvey can identify whether sign errors come from weak negative-number skills, equation solving or number-line notation. Personalised questions, immediate feedback and exam-style practice help students work at an appropriate pace and build confidence.

Contact and lesson enquiry

Students and parents can contact MasterMaths Tutoring to discuss personalised GCSE, IGCSE or KS3 support.