Solving Inequalities

How do you solve linear inequalities?

Solving a linear inequality follows the same balance method as solving equations. Add, subtract, multiply or divide both sides to isolate x. The one crucial difference is the golden rule:

Multiply or divide by a negative → FLIP the inequality sign

For example, −3x > 12 becomes x < −4 (dividing by −3 flips > to <).

Number line representation

After solving, you can represent the solution on a number line:

• Open circle (&cir;): Used for < or > — the value is not included.
• Filled circle (•): Used for ≤ or ≥ — the value is included.

Draw a line or arrow from the circle in the direction of the solution set. For a double inequality such as −2 < x ≤ 5, mark open at −2, filled at 5, and shade between.

How do you solve quadratic inequalities?

Quadratic inequalities are a Higher tier topic. The most reliable approach is the critical values method:

1. Step 1 — Solve as an equation: Replace the inequality sign with = and solve (usually by factorisation) to find the critical values (roots).
2. Step 2 — Sketch the curve: Draw a U-shaped parabola passing through the critical values on the x-axis. A positive x² coefficient gives a U shape; negative gives an inverted U.
3. Step 3 — Identify the region: For > 0 or ≥ 0, the solution is where the curve is above the x-axis (outside the roots). For < 0 or ≤ 0, the solution is where the curve is below the x-axis (between the roots).

Write the final answer using set notation or inequality notation. For "outside the roots" solutions, you need an OR statement: x ≤ a or x ≥ b. For "between the roots", use a combined inequality: a ≤ x ≤ b.

How do you identify graphical inequality regions?

Graphical inequality questions ask you to shade a region on a coordinate grid that satisfies one or more inequalities. Two key rules for the boundary lines:

• Solid line: ≤ or ≥ — points on the line are included in the region.
• Dashed line: < or > — points on the line are not included.

To decide which side of a boundary line to shade, pick a test point (often the origin (0, 0) if it is not on the line). Substitute into the inequality — if true, shade that side; if false, shade the opposite side. When multiple inequalities are combined, the solution is the region where all shading overlaps.

Worked Examples

Common Mistakes

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