Linear inequalities as regions

A line divides the plane into two half-planes. Draw its boundary by replacing the inequality sign with equality. Use a broken boundary for a strict inequality and a solid boundary when equality is included. Test a point away from the line to choose the correct side.

Example

For \(x\ge1\), keep the side to the right of the solid vertical line. For \(x+y<4\), the point \((0,0)\) satisfies the inequality, so keep that side of the broken line \(x+y=4\).

For simultaneous conditions, keep only their common region. Cambridge questions normally shade unwanted regions unless directed otherwise. The illustration instead shades the permitted region \(R\) for \(x\ge1,\ y\ge0,\ x+y<4\).

The blue region does not include the broken sloping boundary. A point must satisfy every inequality; being on the correct side of just one boundary is insufficient.