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An easy way to approach this ineq will be to analyze that: |x-z| + |x| = |z| means |z-x| = |z| - |x|. |z-x| is the distance between z and x on number line. It can only be equal to |z| - |x| if both z and x have the same signs.

a) z < x - implies that y > 0 because zy < xy. If y > 0 then z < x < 0. Therefore both have same signs. SUFF

b) y>0 then z < x < 0. Therefore both have same signs. SUFF

This question was discussed before here is my post from there:

This is not a good question, as neither of statement is needed to answer the question, stem is enough to do so.

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.

Re: If zy < xy < 0, is |x-z| + |x| = |z|? [#permalink]

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17 Dec 2013, 03:17

I solved this question on number line, we can first analyze this eq |x-z| + |x| = |z|? This equality is only possible, if x and z are on the same side of the number line.

(-) ----z----x----0--- or ---0-----x----z---- (+)

|distance between xandz| + |distance of x from origin| = |distance of z from origin|

given zy<xy<0 which can happen in two case.

z x y - - + zy negative xy negative < both less than 0 + + - zy negative xy negative < both less than 0

Therefore we don't even need option 1 and 2 to validate this |x-z| + |x| = |z|.

Answer : D
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Piyush K ----------------------- Our greatest weakness lies in giving up. The most certain way to succeed is to try just one more time. ― Thomas A. Edison Don't forget to press--> Kudos My Articles: 1. WOULD: when to use?| 2. All GMATPrep RCs (New) Tip: Before exam a week earlier don't forget to exhaust all gmatprep problems specially for "sentence correction".

Re: If zy < xy < 0, is |x-z| + |x| = |z|? [#permalink]

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10 May 2014, 11:04

1

This post received KUDOS

using Statement 1 : from the question , zy < xy y(z-x)<0 ------- (A) Now statement 1 tells me that (z-x)< 0 . This implies Y>0 So, if zy < xy < 0 and Y>0 This implies Z & X < 0 mod (x-z) + mod (x) = X-Z (since Z-X<0) + (-X) = -Z = |Z|

Using Statement 2 : From (A), y(z-x)<0 Since from Statement 2 we know that y > 0 That implies (z-x)< 0 .

Re: If zy < xy < 0, is |x-z| + |x| = |z|? [#permalink]

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16 May 2015, 06:49

Hello from the GMAT Club BumpBot!

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Re: If zy < xy < 0, is |x-z| + |x| = |z|? [#permalink]

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19 Jun 2016, 00:46

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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