Prove that a certain set is a halfplane.

Question

Let $a,b,c \in \mathbb{R}$ with $(a,b)\ne0$. Prove that the set $S=\{(x,y)\in \mathbb{R}^2: ax+by+c\le0\}$ is a halfplane of $\mathbb{R}^2$.

I'm a bit confused by this exercise, For what I know the definition of halfplane is what I'm asked to prove (the book does not give a definition of halfplane), searching on the web I found:

A half-plane is a planar region consisting of all points on one side of an infinite straight line, and no points on the other side. (Wolfram Alpha)

Either half of an infinite plane (divided by a straight line). (Wiktionary)

...and other similar definitions; but I don't know how to prove the assertion using these.

Can you give me a hint? Thanks

Answer 1

A half plane in $ \mathbb R^2$ is defined as follows: given a linear functional $f: \mathbb R^2 \to \mathbb R$ and $d \in \mathbb R$, the set

$H=\{(x,y) \in \mathbb R^2: f(x,y) \le d\}$

is called a half plane.

For your set $S$ take $f(x,y)=ax+by$ and $d=-c$.

Answer 2

If $b>0$, then $S$ is the same as the set of all $(x,y)$ with $y\le (-a/b)x+(-c/a)$, that is the set of points below the line $y=(-a/b)x+(-c/b)$.

Likewise, if $b<0$ then $S$ is the set of points above some line.

If $b=0$, then $S$ is the set of points to the right of, or to the left of, some vertical line.