Linear Inequations in two variables

A straight line $ax+by+c=0$ (with $a,b$ not both zero) divides the whole coordinate plane into three pieces: the points

lying exactly on the line, and two separate half-planes on either side of it. These two half-planes are described by the

strict inequalities $ax+by+c<0$ and $ax+by+c>0$ — every point of the plane satisfies exactly one of these two, or lies on

the line itself. When the boundary line is included as well, we get the two CLOSED half-planes $\{(x,y): ax+by+c\le0\}$ and

$\{(x,y): ax+by+c\ge0\}$, whose common boundary is the line itself, $\{(x,y): ax+by+c=0\}$.

A linear inequation in two variables $x,y$ is a mathematical expression of the form $ax+by<c$ or $ax+by>c$ (or with

$\le,\ge$), where $a\ne0$ and $b\ne0$ are not simultaneously zero, and $a,b,c\in\mathbb{R}$. Graphically, such an

inequation names exactly one of the two half-planes cut out by the line $ax+by=c$: which side depends on the direction of

the inequality symbol.

Worked Activity — checking points against $2x+3y-6\le0$. Five ordered pairs are tested by substituting each into the

left-hand side $2x+3y-6$ and comparing the result with $0$:

• $(1,-1)$: $2(1)+3(-1)-6=2-3-6=-7$, and $-7\le0$, so this point IS a solution.
• $(2,1)$: $2(2)+3(1)-6=4+3-6=1$, and $1\le0$ is FALSE, so this point is NOT a solution.
• $(-2,1)$: $2(-2)+3(1)-6=-4+3-6=-7$, and $-7\le0$, so this point IS a solution.
• $(-1,-2)$: $2(-1)+3(-2)-6=-2-6-6=-14$, and $-14\le0$, so this point IS a solution.
• $(-3,4)$: $2(-3)+3(4)-6=-6+12-6=0$, and $0\le0$ (equality holds), so this point IS a solution — it lies exactly on the boundary line $2x+3y-6=0$, which the closed inequality $\le0$ still includes.

So four of the five points ($(1,-1)$, $(-2,1)$, $(-1,-2)$, $(-3,4)$) solve the inequation, and only $(2,1)$ fails it — this

kind of direct substitution-and-compare check is the most basic way to test whether a single point belongs to a linear

inequation's solution set, before moving on to describing the FULL solution set graphically (the whole shaded half-plane)

in the sub-sections that follow.