If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

This is not a good question, (well at least strange enough) as neither of statement is needed to answer the question, stem is enough to do so. This is the only question from the official source where the statements aren't needed to answer the question. I doubt that such a question will occur on real test but if it ever happen then the answer should be D.

If zy<xy<0 is |x-z|+|x| = |z|

Look at the inequality zy<xy<0:

We can have two cases:

A. If y<0 --> when reducing we should flip signs and we'll get: z>x>0.
In this case: as z>x --> |x-z|=-x+z; as x>0 and z>0 --> |x|=x and |z|=z.

Hence in this case |x-z|+|x|=|z| will expand as follows: -x+z+x=z --> 0=0, which is true.

And:

B. If y>0 --> when reducing we'll get: z<x<0.
In this case: as z<x --> |x-z|=x-z; as x<0 and z<0 --> |x|=-x and |z|=-z.

Hence in this case |x-z|+|x|=|z| will expand as follows: x-z-x=-z --> 0=0, which is true.


So knowing that zy<xy<0 is true, we can conclude that |x-z|+|x| = |z| will also be true. Answer should be D even not considering the statements themselves.

As for the statements:

Statement (1) says that z<x, hence we have case B.

Statement (2) says that y>0, again we have case B.

Answer: D.

Hope it helps.
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If zy < xy < 0, is | x - z | + |x| = |z|?

1) z < x
2) y > 0
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You can solve such questions easily by re-stating '< 0' as 'negative' and '> 0' as 'positive'.

zy < xy < 0 implies both 'zy' and 'xy' are negative and zy is more negative i.e. has greater absolute value as compared to xy. Since y will be equal in both, z will have a greater absolute value as compared to x.

When will zy and xy both be negative? In 2 cases:
Case 1: When y is positive and z and x are both negative.
Case 2: When y is negative and z and x are both positive.

Question:
Is | x - z | + |x| = |z| ?
Is | x - z | = |z| - |x| ?
Is | z - x | = |z| - |x| ? (Since | x - z | = | z - x |)
We re-write the question only for better understanding.

Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign. In case both have the same sign, they get subtracted on both sides so you get the same answer. In case they have opposite signs, they get added on LHS and subtracted on RHS and hence the equality doesn't hold.

So if we can figure whether both x and z have the same sign, we can answer the question.

As we saw above, in both case 1 and case 2, x and z must have the same sign. This implies that the equality must hold and you don't actually need the statements to answer the question. You can answer it without the statements (this shouldn't happen in actual GMAT).
Hence answer must be (D).
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Since zy < 0 and xy < 0, both z and x have opposite sign to y, so they must be either both positive or both negative. In other words, we know that xz > 0.

(1) Given that z < x, when both z and x are negative, |z - x| + |x| = -z + x + (-x) = -z = |z| TRUE
z and x cannot be both positive, because then y would be negative, and from zy < xy we would obtain that z > x.
Sufficient.

(2) Knowing that y > 0, we can deduce that both z and x are negative. In addition, from zy < xy it follows that z < x, and we are in the same case as above.
Sufficient.

Answer D
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If | z - x |= |z| - |x| assuming x & z have same sign
| x - z |= |z| - |x| ; Since | x - z | = | z - x |

Implies |z| - |x| = |x| - |z|
which is defn not true
z = 1 , x = 2
Plugging, we get
- 1 = 1 ?!?

Have I misunderstood something?
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The question is not good.
The target question "If zy < xy < 0, is | x - z | + |x| = |z|?"
tells us zy<xy<0, that is, z and x have the same sign,
Thus, target question could be rephrased as "| x - z | = |z|-|x| ?"=> "|z|>|x| ?"
The condition zy<xy<0 could be rephrased as |zy|>|xy|>0 => |z|>|x| , which is already enough to solve the question.