How do I check that a point lies in the region of the acute angle between two intersecting lines?
For example, how do I prove that the point $(2,4)$ lies in between the lines:
$$x - y + 3=0$$ $$x - 2y + 4=0$$
I tried to plot the lines and the point on a graph, and I can confirm the result by graphing:
But I want a method that can be used in an exam as it takes a lot of time to do.
You don't actually need to plot the graph.
The point where the two lines meet is $(-2,1)$ by elimination. Since $(2,4)$ has a greater $x$-coordinate than $(-2,1)$, the region you are looking for is $x - y + 3 > 0$, $x-2y+4 < 0$ (a quick, generic sketch helps).
Therefore you just need to substitute $(2,4)$ into the inequalities and check if they are true.