so, I mostly figured it out, but hopefully somebody can clarify something for me...

is always even due to the 2

so, when x is even, is even and will be an even plus an even, so is even which means that w is even.

when x is odd, is odd and will be an odd plus an even, so is odd which means that w is odd.

So x and w are both even or x and w are both odd, so w - x and w + x will always be even.

The equation becomes

So I divide through by two, apparently just because I can since they are all even (is there another reason?) which gives me

Using an earlier theorem, i can say that

doing the math, that gives me and and this leads to the further result that

This result is correct according to the back of the book except for two things...

first, apparently since it wants all solutions, I multiply all my answers by .

second, the book puts absolute value signs around the entire x answer. The actual answers are:

The second problem I did, , I did in much the same way except it does matter what y is in this case, so I end up not dividing through by 2. I end up with the equations and I come up with the correct answer, but the answer for x is again enclosed with absolute value signs. The same answers result for the third equation of .

So where are the absolute value signs coming from? I barely understand the problems as it is and the book of course does not mention absolute values in the section.