During my read through the Elements of Coordinate Geometry by S.L Lonely, I've stumbled upon his method of proving that one point is either on one side or the other of the graph of a straight line based on the sign of the answer you get when you substitute the coordinates of the said point in the equation to the given straight line. Namely that some point $P(x_1,y_1)$ is on one side or the other of a given straight line $Ax + By + C = 0$ based on whether $Ax_1 + By_1 + C$ is positive or negative. The book later goes on to finding the cases in which a point might be on the same side of a straight line with the origin or that two points might be on the same side of a straight line. But I didn't quite grasp his method of proof. I used a different way to prove this same result, but would still like to know what S.L Lonely is conveying in his book. Here is the section so talked about:
To be specific, I don't understand what how he concludes this part here:
It is clear from the figure that PQ is drawn parallel to the positive or negative direction of the axis of y according as P is on one side, or the other, of the straight line LM, i.e. according as y" is > or < y', i.e. according as y" -y' is positive or negative. Now, by (1 ), y" — y' = - (Ax +C)÷B The point (x', y' ) is therefore on one side or the other of LM according as the quantity Ax + By + C is negative or positive.