Graphing Inequalities in Two Variables When we have to graph an inequality in two variables, we usually graph the corresponding equality, i.e. the straight line on the coordinate plane, which divides the plane into two parts. Then, we use a so-called 'test point' to decide which of the two parts represents the inequality.
So, here we clearly assume that one of the two parts formed by the straight line represents the '>' inequality, whereas the other part represents the '<' inequality.
Is there any rigorous proof that can show that this is indeed the case? Thanks.

The assumption you talk about clearly follows from the trichotomy law for real numbers.
http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29
