IMO : C
(1) x+y=z - insufficient, plugging values x=4,y=2,z=6,we get 4+2=6
absolute value(4)=4=6-2....correct,
but if we take x<0 ,say x=-4, y=2,z=-2,we get -4+2=-2
Absolute value of(-4) = 4 = -2-2=-4...incorrect
(2) gives us x<0, which is the case 2 of explanation above. this is not sufficient alone but sufficient when (1) &(2) are taken together.

Earth@work u r right about C) but may be not on the values picked

Stmt I

but if we take x<0 ,say x=-4, y=2,z=-2,we get -4+2=-2
Absolute value of(-4) = 4 = -2-2=-4...incorrect

|-4|= 4 y-z = 2-(-2) = 4 so |x| = y-z

x=1 y=2 z=3

|X| = 1 but y-z = -1 so |x| <> y-z

INSUFF

Stmt II

x<0

x=-10 y=-100 z=-1000

|x| <> y-z

x=-1 y=2 z=1

|x| = y-z

INSUFF


Stmt I and II together

X+Y = Z
X<0

Take y negative(y>z) or positive(y>z) |x| = y-z

C)

Folks,

I was thinking in entirely different way for this question..

Can anyone let me know the issue with my approach...

Question asked: Is |x| = Y - Z

this can be re-written as X = Y - Z and X = Z - Y or 2Z = 2Y

or Question asked can be re-written as is Z = Y?

From 1, we know that X + Y = Z ; we dont know the individual values...If X = 0, then Y = Z..if not, then Y != Z...So option A not sufficient..

From 2, we can derive nothing about Y and Z...

Combining both 1 and 2, since we know that X < 0, Y cannot be equal to Z...

So C is the answer.

I am taking little different approach, would appreciate if anyone can let me know any issue with this approach ?

Question stem
Is |x| = y-z
==> we need to check
if y-z = +ve
From Statement 1
x+y = z ==> y - z = -x
thus y-z will be +ve if x is -ve hence insuff.

From Statement 2:
x<0 again insufficient

From 1 and 2 we get x<0
hence sufficient.

Looks good to me Amit! Nice work.