Represent two-variable inequalities as regions

By UK GCSE Mathematics · independent study · myAudience

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Represent two-variable inequalities as regions

A two-variable inequality describes points in a region. Its boundary is the corresponding equation. Use a solid boundary where equality is included and a dashed boundary where it is excluded.

Test a point away from the boundary to choose the correct side. For y > x + 1, the point (0, 0) fails, so shade the opposite side of the line. Multiple inequalities require the intersection of their permitted regions.

An answer may have no feasible region, a bounded region or an unbounded region. Keep the boundaries and inclusion rules visible. A shading convention that marks the unwanted region can also work, but it must be stated consistently.

Worked example: Describe the region x ≥ 0, y ≥ 0, x + y ≤ 4.

Use the coordinate axes and x + y = 4 as solid boundaries.

Keep the first quadrant.

Keep the points at or below the line.

Answer: The closed triangular region with vertices (0, 0), (4, 0), (0, 4).

Check your understanding: Is (3, 2) in this region?

Answer: No.

Why: Although both coordinates are nonnegative, their sum 5 exceeds 4.