Alternative Proof to Proving Inequality Involving Slopes?

Question

I was wondering if there was any other way of proving:

Is this idea regarding slope true and how do you prove it?

As I feel like I wouldn't be to catch that on say, a test, I was hoping if there is something else that would work in proving this.

Answer 1

Let $AB$ lie on the $x$-axis in a new coordinate system. It's obvious that we can do that by translation and rotation, which doesn't change the slopes of any of the lines. Now the slope of $AB$ is $0$. From the condition we have that the slope of $BC$ is positive, so $C$ must lie above the $x$-axis, i.e. $C_y$ (the y coordinate of $C$) $>0$. But as $A_x < B_x$ the problem reduces to proving that the segment $AC$ lies above the segment $AB$, which is obvuously true. To make it more rigorous you can write that:

$$\text{slope of } AC = \frac{C_y - A_y}{C_x - C_y} = \frac{C_y}{C_x - C_y} > 0 = \text{slope of } AB$$

I don't think that this is that hard to notice during an exam.