Numerical inequalities

The essentials

An inequality compares two quantities using < (less than), ≤ (less than or equal to), > (greater than) or ≥ (greater than or equal to). Many real situations are naturally inequalities: a budget that must not be exceeded, a minimum age or pass mark, or the capacity of a lift.

Translating words: "at least" and "not less than" mean ≥; "at most", "not more than" and "maximum" mean ≤; "more than" means >; "less than" means <.

Linear inequalities are solved like equations with one important rule: when you multiply or divide both sides by a negative number, the inequality sign reverses. Adding or subtracting the same number, or multiplying by a positive number, keeps the sign.

Key points

• Example: 3x − 5 < 7 gives 3x < 12, so x < 4.
• Example: −2x ≥ 6 gives x ≤ −3 (sign reversed).
• |x − a| < b (with b > 0) means a − b < x < a + b.
• |x − a| > b means x < a − b or x > a + b.
• Solutions can be shown on a number line; a filled dot means the end point is included (≤ or ≥) and an open dot means it is excluded.
• For averages: "average of n marks at least m" means the total is at least nm.

Exam tips

• Underline the key phrase ("at least", "not more than") before writing the inequality.
• Check your answer by substituting a value from the solution set.

Common mistakes

• Forgetting to reverse the sign when dividing by a negative number.
• Using < when the situation allows equality (≤).