Regions defined by linear inequalities revision notes

Edexcel International GCSE Maths (4MA1), chapter 8: Inequalities, specification point 2.8b. Free lesson.

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A region defined by linear inequalities is the part of a grid where all the inequalities are true at once. Draw each boundary correctly, choose the side that satisfies it, then shade and label the overlap.

How to draw each boundary

Turn each inequality into an equation to get its boundary line. The inequality symbol tells you whether points on that line are included, so it decides whether the line is solid or dashed. Use a dashed line for a strict inequality, < or >, because the boundary is not included. Use a solid line for ≤ or ≥, because points on the boundary are included. For example, x > 1 has the dashed boundary x = 1. The inequality does not include x = 1.

How to choose which side to shade

A boundary line divides the grid into two halves. Test the origin, (0, 0), in the inequality. If it makes the inequality true, keep the side containing the origin. If it is false, keep the other side. For x > 1, the origin gives 0 > 1, which is false, so keep the side away from the origin. For y ≤ 4, the origin gives 0 ≤ 4, which is true, so keep the side containing the origin.

How to show the region for two inequalities

Each inequality keeps one half of the grid. The answer is the part that satisfies both conditions, so shade where the two chosen halves overlap and label it R. For x > 1 and y ≤ 4, shade to the right of x = 1 and on or below y = 4. The line x = 1 is dashed, while y = 4 is solid.

How to find an overlap with three inequalities

For three conditions, each boundary again keeps one half of the grid. If a boundary is sloping, rearrange the inequality into a line you can draw. The region R is the overlap of all three chosen halves. For y ≥ 1, keep the area above y = 1. The origin fails the inequality. For y ≤ x + 2, keep the area below the line y = x + 2. The origin satisfies the inequality. For x + y ≤ 6, the origin also satisfies the inequality, so keep its side of the line x + y = 6. All three boundaries are solid because each inequality includes equality.

How to check a point and finish the grid

To check whether a point is inside a region, substitute its coordinates into the inequality. For a question asking you to show a region on the grid, the shaded picture is the answer, not just the inequalities written beside it. For y ≥ x, the boundary y = x is solid. At (1, 4), substitute x = 1 and y = 4: 4 ≥ 1 is true, so the point is inside. Shade the overlap and write R inside it.