I received a PM asking for comment.

uslalas22 wrote: Since statement 1 says X+Z is not equal Y, it tells us X is definitely negative, (or simply any arbitrary number.)

Your line of reasoning here wasn't entirely clear to me, but Statement 1 does not guarantee that X is negative - X+Z can be different from Y when X is positive as well. It is, however, true that we always get a 'no' answer to the question when X is positive (I go into more detail below), so if we want to get a 'yes' answer to the question, we'll need to look at negative values of x.

djiddish98 wrote:

If |X| = Y - Z, then y > z, since the answer has to be positive (or y = z and x = 0)

x = y - z when x is positive
-x = z - y when x is negative

both cases let us know that x + z = y when we rearrange terms. Since statement 1 tells us that x + z != y, then we know that |X| != Y - Z, which is sufficient.

There's a problem with your math here, which I've highlighted in red. If you are considering the case when X is negative, and you have the equation |X| = Y-X, then since X is negative you can replace |X| with -X. But you certainly should not replace Y-Z with Z-Y; Y-Z is still equal to Y-Z. That is, you should not apply a negative sign to both sides of the equation, as you did above; those will always just cancel each other out no matter what equation you look at. You want only to replace |X| with -X, nothing more. If it's not clear why that's the case, imagine a simpler equation: |x| = 3. When you want to find the negative solution for x, when you are getting rid of the absolute value brackets you don't put a negative sign on both sides of the equation - you only put a negative on the left side.

sourabh33 wrote:Is |X| = Y-Z ?

1. X+Z is not equal Y
2. X<0

If, from Statement 1, X+Z is not equal to Y, then certainly X is not equal to Y-Z. If X is not equal to Y-Z, and if X is positive, then |X| will also not equal Y-Z, so if X is positive, the answer to the question is 'no'. But we need to consider the possibility that X is negative as well - this is how we might get a 'yes' answer to the question. If, say, X = -1, Y = 2 and Z = 1, then Statement 1 is true (X+Z is not equal to Y) and |X| = Y-Z is also true - we can get a 'yes' answer to the question as well. The answer is E here, since it's also possible to get a 'no' answer when X is negative (just use the same numbers as before but let X be -10, say).