Math: simplifying

Question

Simplify, if known x<y

(2|x-y|-|y-x|)/(x^2 + y^2 -2xy)

Answer 1

Hi! Well, that's a tricky one!,

First, let's simplify the numerator.

The absolute value of x-y equals the absolute value of y-x, because x-y = - (y-x). Try a few examples of values for x and y to prove it to yourself if the algebra doesn't make sense to you.

That means that we can just subtract and the numerator becomes |x-y|

The denominator, when rearranged, becomes x^2 -2xy + y^2. This factors to (x-y)^2, and the fraction becomes

|x-y|/(x-y)^2

The value of (x-y) divides out, and the answer is just 1/{x-y). But we have a small problem. The fraction above would give us a positive value, and the reduced one will give us a negative one if x<y. We can use an absolute value sign to make the reduced answer true:

1/|x-y|

We could have also called the answer 1/(y-x); we'd get the same value either way.

Hope this helps you!