Technique

A linear inequality in two variables is a half-plane

Algebra • Expressions • Inequalities

Statement

The solutions of

a x + b y \geq c

are the points on one side of the line

a x + b y = c

, together with the line itself when the inequality is not strict; substitute any point off the line - the origin, whenever it is not on it - and keep the side that satisfies the inequality. The solution set of several such inequalities is the intersection of their half-planes.

How to use it

1Draw the boundary from its intercepts: $a x + b y = c$ meets the axes at $(\frac{c}{a}, 0)$ and $(0, \frac{c}{b})$ .
2Test the origin whenever it is off the line, because $0 \geq c$ is a one-glance check.
3To decide whether a whole quadrant is reached, assume the signs that define it and see whether the inequalities then contradict one another - that settles the quadrant with no drawing at all.

Worked example

Which side of the line

2 x + y = 4

satisfies

2 x + y \leq 4

, and does that region contain

(3, 1)

Step-by-step

1
$2 (0) + 0 = 0 \leq 4$
Why: The origin is not on the line, so it may be used as the test point, and it satisfies the inequality.
2
${(x, y) : 2 x + y \leq 4}$ is the side of $2 x + y = 4$ containing the origin, boundary included
Why: The test point decides the side; the sign $\leq$ keeps the line.
3
$2 (3) + 1 = 7 > 4$
Why: $(3, 1)$ fails the inequality, so it lies on the far side.

Answer

The origin's side, with the line itself;

(3, 1)

is not in the region.

Watch out

Never choose a test point that lies on the boundary: it satisfies the equality and tells you nothing about either side. And multiplying or dividing an inequality by a negative number reverses it, so fix the form before reading off a side.

Also known as

half plane, graph of a linear inequality, shaded region, feasible region, system of linear inequalities, test the origin, linear inequality in two variables

Appears in past papers

2018 (II)

Q48