Graphical representation of linear inequations in two variables

This sub-section fixes the standard shading convention for four increasingly general families of linear inequation in two

variables, each illustrated with a small reference graph.

1) $x\le h$ and $x\ge h$. Draw the vertical line $x=h$ in the $XOY$ plane. Every point to the LEFT of this line has an

$x$-coordinate smaller than $h$, so the solution set of $x\le h$ is the closed half-plane on and to the left of the line;

correspondingly, the solution set of $x\ge h$ is the closed half-plane on and to the right of the line.

2) $y\le k$ and $y\ge k$. Draw the horizontal line $y=k$. The solution set of $y\le k$ is the closed half-plane on and

below this line, and the solution set of $y\ge k$ is the closed half-plane on and above it.

3) $ax+by\le0$ and $ax+by\ge0$ (line through the origin). When the line $ax+by=0$ passes through the origin (no

constant term), the origin lies exactly ON the boundary and so cannot be used to test which side is which — a different

point not on the line must be substituted instead, and the half-plane containing that test point is shaded according to

whichever inequality it satisfies.

4) $ax+by\le c$ and $ax+by\ge c$ (general line, using the origin as test point). Draw the line $ax+by=c$; this line

divides the plane into two half-planes, conventionally labelled $H_1$, the "origin side" of the line (the half-plane

containing the point $(0,0)$), and $H_2$, the "non-origin side" (the other half-plane). Substitute the origin's

coordinates into the given inequation: if $(0,0)$ satisfies $ax+by\le c$, then the required region is the origin side

$H_1$, and $H_1$ should be shaded; otherwise, the required region is the non-origin side $H_2$. The points where …