Inequalities
Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations which have specific solutions, inequalities describe ranges of values that satisfy the given condition. The fundamental rule when solving inequalities is that multiplying or dividing both sides by a negative number flips the inequality sign.
Why it matters
Inequalities model countless real-world constraints and boundaries. A shopkeeper might need at least £500 in daily sales to break even, represented as s ≥ 500. Speed limits create inequalities — driving at 35 mph in a 30 mph zone violates v ≤ 30. Manufacturing tolerances use inequalities: a bolt diameter might need to satisfy 9.8 ≤ d ≤ 10.2 millimetres. In GCSE mathematics, inequalities appear across multiple topics including linear programming, quadratic functions, and graphical work. Students encounter them in Year 8 with simple linear inequalities on number lines, progressing to quadratic inequalities in Year 11. Understanding inequalities is essential for A-level mathematics, particularly in calculus when finding intervals where functions increase or decrease, and in statistics when working with confidence intervals and hypothesis testing.
How to solve inequalities
Inequalities
- Solve like an equation (same operations on both sides).
- If you multiply or divide by a negative, FLIP the sign.
- Graph on a number line (open circle for < >, closed for ≤ ≥).
Example: -2x > 6 → x < -3 (sign flipped).
Worked examples
x + 4 < 12
Answer: x < 8
- Understand the problem → x + 4 < 12 — This is like an equation, but instead of '=' we have '<'. We solve it the same way.
- Subtract 4 from both sides → x + 4 − 4 < 12 − 4 → x < 8 — Isolate x by removing the constant from the left side.
- Check with a test value → Try x = 7: 7 + 4 = 11 < 12 ✓ — Pick a value of x that satisfies x < 8 and verify it works in the original inequality.
5x + 8 < -7
Answer: x < -3
- Write the inequality → 5x + 8 < -7 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 8 from both sides → 5x + 8 − 8 < -7 − 8 → 5x < -15 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 5 → x < -3 — Divide by 5 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
- Verify with a test value → Try x = -4: 5·-4 + 8 = -20 + 8 = -12 < -7? ✓ — Pick x = -4 (which satisfies x < -3) and check it works in the original inequality.
4x + 5 ≥ 1
Answer: x ≥ -1
- Write the inequality → 4x + 5 ≥ 1 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
- Subtract 5 from both sides → 4x + 5 − 5 ≥ 1 − 5 → 4x ≥ -4 — Remove the constant term from the left side. The inequality sign stays the same.
- Divide both sides by 4 → x ≥ -1 — Divide by 4 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
- Verify with a test value → Try x = 0: 4·0 + 5 = 0 + 5 = 5 ≥ 1? ✓ — Pick x = 0 (which satisfies x ≥ -1) and check it works in the original inequality.
Common mistakes
- Forgetting to flip the inequality sign when dividing by -2 in -2x > 6, writing x > -3 instead of x < -3
- Using the wrong circle type on number lines — drawing a closed circle for x < 5 instead of an open circle
- Incorrectly combining inequalities like 3 < x < 8 and x > 10, claiming solutions exist when the intersection is empty