Inequalities

This lesson covers linear and quadratic inequalities, representing solutions on number lines and using set notation. The Edexcel 9MA0 specification requires you to solve inequalities algebraically and to interpret them graphically.

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Linear Inequalities

Linear inequalities are solved in the same way as linear equations, with one critical rule: if 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

Example: Solve 4 − 2x ≥ 10.

• −2x ≥ 6
• x ≤ −3 (inequality reversed because we divided by −2)

Double Inequalities

Solve both parts simultaneously.

Example: Solve −3 < 2x + 1 ≤ 7.

• Subtract 1: −4 < 2x ≤ 6
• Divide by 2: −2 < x ≤ 3

This means x is strictly greater than −2 and less than or equal to 3, written as −2 < x ≤ 3 or in set notation {x ∈ ℝ : −2 < x ≤ 3}.

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Quadratic Inequalities

To solve a quadratic inequality: