Linear Inequations in Two Variables — Graphical Solution (Half-Planes)

A linear inequation in two variables has the form $ax+by<c$ (or with $>,\le,\ge$), where $a,b$ are not both zero. Its solution is not a set of isolated points but an entire region of the coordinate plane — a half-plane.

The idea of a half-plane

The corresponding equation $ax+by=c$ is a straight line. That line splits the whole plane into two parts (half-planes), one on each side. Every point on one side satisfies $ax+by<c$, and every point on the other side satisfies $ax+by>c$; points on the line satisfy $ax+by=c$ exactly.

Graphical solution of a two-variable linear inequation

The solution set of $ax+by\ \square\ c$ (where $\square$ is one of $<,>,\le,\ge$) is one of the two half-planes determined by the boundary line $ax+by=c$, together with the line itself when the sign is $\le$ or $\ge$.

The method, step by step