Regions

Regions on a graph represent the areas that satisfy a given inequality or set of inequalities. The boundary of the region is typically a line or curve, which may or may not be included in the solution, depending on whether the inequality includes an equality (e.g., \(\leqslant or \geqslant\)).

Sketching Regions for Linear Inequalities

\[y \leqslant 2x + 3\]

To graph the boundary line, start by converting the inequality into an equation by replacing the inequality sign with an equal sign.

\[y = 2x + 3\]

We now need to work out the appropriate region to shade. We can test a point not on the line to see if its included in the inequality. Usually \((0, 0)\) works well.

For our graph, at \((0, 0)\) we find that \(0 \leqslant 2\times 0 + 3\) so the origin is inside the region that we should shade in.

Sketching Regions for Quadratic Inequalities

For non-linear inequalities, the boundary is a curve. The process is similar to linear inequalities, but the region can be inside or outside the parabola.