Algebraic Inequalities

This lesson covers algebraic inequalities as required by the AQA A-Level Mathematics specification (7357). You must be able to solve linear, quadratic, and rational inequalities, express solutions using set notation and interval notation, and represent solutions on a number line. Inequalities appear throughout pure mathematics and are essential for finding domains, ranges, and conditions for convergence.

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Linear Inequalities

A linear inequality involves a linear expression and is solved in the same way as a linear equation, with one crucial difference:

When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Example: Solve 3x − 7 > 5.

3x > 12
x > 4

Solution: x > 4 or in set notation {x : x > 4} or interval notation (4, ∞).

Example: Solve 2 − 5x ≤ 17.

−5x ≤ 15
x ≥ −3       (inequality reversed because we divided by −5)

Double Inequalities

Example: Solve −3 < 2x + 1 ≤ 9.

−3 < 2x + 1 ≤ 9
−4 < 2x ≤ 8
−2 < x ≤ 4

Solution: {x : −2 < x ≤ 4} or (−2, 4].