Linear Inequalities in Two Variables

Instead of a line, an inequality like y ≥ 2x + 1 describes a whole region of the graph. This chapter teaches you to draw that region, combine several inequalities into one feasible area, and read real constraints, budgets, limits, minimums, into mathematical form.

Paper 2 typically asks you to draw two or three inequalities and identify the region satisfying all of them, sometimes from a real constraint like a shop’s stock limits. Neat lines and a clearly labelled region earn the marks; a rushed graph loses them.

Once you have drawn a boundary line, deciding which side satisfies the inequality does not need any intuition, one test point settles it. Pick a point that is clearly on one side and not on the line itself; the origin (0, 0) is easiest because the arithmetic is trivial, as long as the line does not pass through it.

Substitute the point's coordinates into the inequality. If the statement comes out true, shade the side that contains your test point; if it comes out false, shade the other side.

For example, to place y ≥ 2x + 1, test (0, 0): is 0 ≥ 2(0) + 1, that is 0 ≥ 1? That is false, so you shade the side away from the origin.

If the boundary does pass through the origin, just choose another easy point such as (1, 0).

Real-context questions hide the inequalities inside sentences, and the marks start with translating them correctly. Learn the standard phrases: 'at least' and 'not less than' mean ≥; 'at most' and 'not more than' mean ≤; 'more than' means the strict >; 'fewer than' or 'less than' means the strict <.

Watch for comparisons between the two variables too 'the number of chairs is at least twice the number of tables' becomes y ≥ 2x if y is chairs and x is tables. Finally, many contexts carry hidden non-negativity: you cannot have a negative number of items, hours or ringgit, so add x ≥ 0 and y ≥ 0 even when the words never say so.

Missing these silent constraints is one of the most common ways to lose a mark on an otherwise correct region.

The feasible region is the overlap where every inequality is true at once, so the fastest way to check your final shading is to pick any point that sits inside your shaded area and test it against all of the inequalities in turn. If it satisfies every one, your region is in the right place; if even one fails, the shading is wrong somewhere and you should re-check that particular boundary.

Choose a point well away from the edges so a near-miss does not fool you, the middle of the region is ideal. This one habit catches the usual errors (a line shaded on the wrong side, or a constraint forgotten entirely) before you lose the marks, and it costs only a few seconds of substitution.

This example works through defining a region, finding where two boundaries meet, and testing a point, the core skills of the chapter.