statement 1:
we can rephrase this to x = z - y.
there are thus 3 possibilities for the absolute value |x| :
(a) if z - y is positive, then |x| = z - y, and will NOT equal y - z (which is a negative quantity).
(b) if z - y is negative, then |x| = y - z (the opposite of z - y).
(c) if z - y = 0, then |x| equals both y - z and z - y, since each is equal to 0.
TAKEAWAY: when you consider absolute value equations, you'll often do well by considering the different CASES that result from different combinations of signs.
notice that (a) and (b), or (a) and (c), taken together prove that statement 1 is insufficient.

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statement 2:
we don't know anything about y or z, so this statement is insufficient.**
if you must, find cases: say y = 2 and z = 1. if x = -1, then the answer is YES; if x is any negative number other than -1, then the answer is NO.

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together:
if x < 0, then this is case (b) listed above under statement 1.
therefore, the answer to the prompt question is YES.
sufficient.

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**note that, if i were particularly evil, i could craft a statement that doesn't mention all three of x, y, z and yet IS STILL SUFFICIENT.
here's one way i could do that:
(2) y < z
in this case, y - z is negative and therefore CAN'T equal |x| -- no matter what x is -- since |x| must be nonnegative.
so, this statement is a definitive NO, and is thus sufficient even though it doesn't mention x at all.
this is evil, but i see no reason why it wouldn't be on the test.