SPM Mathematics: How to Master Linear Inequalities in Two Variables

The boundary line is solid when the inequality is greater-or-equal or less-or-equal, and dashed when it is strictly greater or less, because a dashed line means the points on it are not included. After drawing the line, test a simple point such as the origin to decide which side satisfies the inequality, then shade that region.

Getting the line style wrong is an easy mark to drop.

The harder marks come from writing the inequalities yourself: at least 20 becomes greater-or-equal to 20, no more than becomes less-or-equal, and two conditions together become a system. List each condition in words first, then convert it, so you do not lose a constraint.

The shaded overlap of all the regions is the answer they want.

Not every boundary line has a slope, an inequality like x > 2 or y ≤ 5 still needs a full line drawn on the same set of axes, just a vertical or horizontal one instead of a diagonal one. For x > 2, the boundary is the vertical line through x = 2, and the shaded region is everything to one side of it, regardless of the y-value; for y ≤ 5, the boundary is horizontal through y = 5, and shading runs above or below it.

Students sometimes try to plot these as if they need two variables to define the line, which only wastes time, one coordinate is enough to place a vertical or horizontal boundary correctly.

A typical question gives two or three inequalities together and asks for the single region that satisfies all of them at once, usually labelled R. The clean way to build this is to shade each inequality's unwanted side lightly with its own light shading or hatching, one inequality at a time, so that the region left completely unshaded, satisfying every inequality simultaneously, becomes visually obvious.

Shading the wanted side directly for each inequality instead often produces overlapping shading that is hard to read back correctly.

Some questions work in reverse, giving you a shaded region and asking you to state the inequalities that define it. Work one boundary line at a time: find its equation from two points or its gradient and intercept, then check whether the shaded region lies above or below that line to decide the direction of the inequality, and finally check whether the line itself is included (solid, so ≤ or ≥) or excluded (dashed, so < or >) before writing the final inequality.