Inequalities

• Symbols: < less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to.
• Number lines: open circle for < or >, closed (filled) circle for ≤ or ≥.
• Solve like an equation, but reverse the sign if you multiply or divide by a negative.
• Regions on a graph: draw each boundary line, then find the side that satisfies the inequality.
• Higher: quadratic inequalities, and harder regions defined by several lines.

Reading and showing inequalities

A double-ended inequality such as 1 < x ≤ 5 means x is greater than 1 and at most 5. On a number line, draw an open circle where the end value is not included (<, >) and a closed circle where it is (≤, ≥), joined by a line.

The one extra rule

Add, subtract, multiply and divide as for an equation. When you multiply or divide by a negative number, reverse the inequality sign. For double inequalities, do the same operation to all three parts.

Representing and identifying regions

Each inequality has a boundary line found by replacing the sign with =. Draw the line, then test a point not on it (the origin is easiest) to see which side satisfies the inequality. The required region satisfies all the inequalities. At Foundation tier you are not required to show whether boundaries are included; the usual convention is a solid line for ≤ or ≥ and a dashed line for < or >.

Find the critical values, then sketch

Solve the related equation to find the critical values, then sketch the parabola. For a positive x² term the curve is ∪-shaped: it is below the x-axis between the roots and above it outside them.