mode inequality

Note that is zy < xy < 0, none of x, y, and z can be equal to zero.
This means y and z are of opposite signs and x and y are also of opposite signs.
So, x and z are of same sign.

If |x - z| + |x| = |z|, then |x - z| = |z - x| = |z| - |x|
So, distance between z and x is equal to |z| - |x|, i.e. the difference of the distance of z from 0 and distance of x from 0 on the number line.

This is possible only when both x and z lies on the same side of 0 on the number line.
Also, as |z - x| must be positive, |z| - |x| must be positive, i.e. |z| must be greater than |x|
So, the problem is basically asking whether z < x < 0 or 0 < x < z

Statement 1: z < x
Now it is possible that either 0 < z < x or z < x < 0
So, statement 1 is not sufficient

Statement 2: y > 0
So, x and z are both negative.
Now it is possible that either x < z < 0 or z < x < 0
So, statement 2 is not sufficient

Both statements together: Only possibility is z < x < 0
This in turn means |x - z| + |x| = |z|
So, both statements together is sufficient

Answer : C

Here is another approach to solve the problem if you find my first post difficult to understand.

From given information we can conclude that x and z are of same sign.

Statement 1: z < x
If z and x both positive, |x - z| + |x| = (x - z) + x = (2x - z) ≠ |z| ---> NO
If z and x both negative, |x - z| + |x| = (x - z) - x = -z = |z| ---> YES

So, statement 1 is not sufficient

Statement 2: y > 0
So, x and z are both negative.
If x = -2 and z = -1, |x - z| + |x| = |-2 + 1| + 2 = 3 ≠ |z| ---> NO
If x = -1 and z = -2, |x - z| + |x| = |-1 + 2| + 1 = 2 = |z| ---> YES
So, statement 2 is not sufficient

Both statements together: z < x < 0
So, |x - z| + |x| = (x - z) - x = -z = |z|
So, both statements together is sufficient

Answer : C

Experts please confirm, something aint right here, in my opinion OA should be D

vipulgoyal wrote:GIVEN ZY < XY < 0
IS | X – Z | + | X | = | Z |
1) Z < X
2) Y > 0

ZY < XY < 0
If Y>0; 0>X>Z
If Y<0; 0<X<Z

Rephrased question: Is |X – Z| = |Z| - |X|?

statement1: z < x
Lets pick values for 0>X>Z
if X=-1 and Z=-2 ; |X – Z| = |Z| - |X|; 1= -1; this can never be true
this means that | X – Z | + | X | = | Z | is wrong

statement is sufficent

statement 2: y > 0
Lets pick values for 0<X<Z
If X=1 and Z=2 ;|X – Z| = |Z| - |X|; 1 = -1; this can never be true
this means that | X – Z | + | X | = | Z | is wrong

statement is sufficent

ans = d

vipulgoyal wrote:GIVEN ZY < XY < 0
Is | X – Z | + | X | = | Z |?
1) Z < X
2) Y > 0

|x-z| = the distance between x and z.
|x| = the distance between x and 0.
|z| = the distance between z and 0.

Constraint in the question stem: zy < xy < 0.
Case 1: If y is POSITIVE, then x and z are NEGATIVE, with z "more negative" than x, so that zy < xy.
Case 2: If y is NEGATIVE, then x and z are POSITIVE, with z "more positive" than x, so that zy < xy.

Case 1:
z<---|x-z|--->x<---|x|--->0..........y
In the number line above, the red portion represents |z|: the distance between z and 0.
|x-z| + |x| is equal to the red portion.
Thus:
|x-z| + |x| = |z|.

Case 2:
y..........0<---|x|--->x<---|x-z|--->z
In the number line above, the red portion represents |z|: the distance between z and 0.
|x-z| + |x| is equal to the red portion.
Thus:
|x-z| + |x| = |z|.

The two statements are IRRELEVANT.
Given that zy < xy < 0, it will ALWAYS be true that |x-z| + |x| = |z|.
Thus:
in statement 1, the answer to the question stem is YES, since it will always be true that |x-z| + |x| = |z|.
In statement 2, the answer to the question stem is YES, since it will always be true that |x-z| + |x| = |z|.

The correct answer is D.

Since the two statements are irrelevant, this problem is flawed.