How do you evaluate absolute value statements such as these?

absolute signs make any sign inside (+ or -) positive (+).

So in your first example

-M=|-M|

we can simplify |-M| to |M|, so:

-M=|M|

any value of M<=0 works:

-(-1) = |-1|
1 = 1


your second example must be 0:
|P|=-|Q|

1 or -1 will not work because the absolute signs turn anything to a positive

|whatever| = +ve whatever so

+ve = -1*+ve

only 0 will make this equation work

Hope this helps

Thanks - definitely makes sense!