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

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Manager
Joined: 17 Aug 2009
Posts: 176
If zy < xy < 0, is |x-z| + |x| = |z|?  [#permalink]

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28 Jan 2010, 01:50
4
18
00:00

Difficulty:

95% (hard)

Question Stats:

39% (01:41) correct 61% (01:41) wrong based on 633 sessions

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

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

For zy < xy < 0 to be true, I am counting two possible scenarios

x y z
-ve +ve -ve -------------------1
+ve -ve +ve--------------------2

Statement 1
rules out scenario 2 but scenario 1 is possible.
Now when i substitute the signs of x and z and take them out from the modulus, i get -

(-x + z) + (-x) = (-z)
-2x = -2z
x=z - - -
Therefore in the original equation, |x-z| = 0
and |x| = |z|

-hence sufficient

Statement 2

Again eliminates the possibility of the scenario 2
and hence is sufficient

Am i correct here?
Manager
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Joined: 21 Nov 2009
Posts: 58
WE: Project Management (Manufacturing)

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28 Jan 2010, 02:51
9
1
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

D is the answer i think
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Math Expert
Joined: 02 Sep 2009
Posts: 52231

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28 Jan 2010, 05:14
6
3
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.

Hope it helps.
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Joined: 31 Jan 2010
Posts: 8

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31 Jan 2010, 11:40
1
wow, I solved this question by myself:)
SVP
Joined: 06 Sep 2013
Posts: 1705
Concentration: Finance
Re: If zy < xy < 0, is |x-z| + |x| = |z|?  [#permalink]

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27 Nov 2013, 07:32
1
zaarathelab wrote:
If zy < xy < 0, is |x-z| + |x| = |z|?

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

For zy < xy < 0 to be true, I am counting two possible scenarios

x y z
-ve +ve -ve -------------------1
+ve -ve +ve--------------------2

Statement 1
rules out scenario 2 but scenario 1 is possible.
Now when i substitute the signs of x and z and take them out from the modulus, i get -

(-x + z) + (-x) = (-z)
-2x = -2z
x=z - - -
Therefore in the original equation, |x-z| = 0
and |x| = |z|

-hence sufficient

Statement 2

Again eliminates the possibility of the scenario 2
and hence is sufficient

Am i correct here?

Rearrange question |-(-x+z)|=|z-x| = |z|-|x|?
By property they will be equal when both x and z have the same sign

Statement 1
If x>z, then with zy < xy < 0, x and z are both positive. Same sign. Suff

Statement 2.
If y>0 then same with zy < xy < 0, x and z both positive. Same sign Suff

Hope it helps
Cheers!
J
Director
Status: Everyone is a leader. Just stop listening to others.
Joined: 22 Mar 2013
Posts: 772
Location: India
GPA: 3.51
WE: Information Technology (Computer Software)
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|.

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

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16 Apr 2014, 21:28
(1) z < x => y > 0
(2) Same as (1)

(1) or (2)
x < 0
z < 0

0 < x-z
abs(x-z) + abs(x) = abs(z)?
x-z + -x = -z
S

D
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Location: United States
Concentration: General Management, Entrepreneurship
GMAT Date: 06-30-2014
GPA: 3.99
WE: Analyst (Consulting)
Re: If zy < xy < 0, is |x-z| + |x| = |z|?  [#permalink]

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10 May 2014, 11:04
1
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 .

Hence D.
Intern
Joined: 18 Jan 2017
Posts: 36
Re: If zy < xy < 0, is |x-z| + |x| = |z|?  [#permalink]

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14 Mar 2017, 22:37
Bunuel wrote:
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.

Hi Bunuel, your explanation is totally convincing.

Can you or someone please confirm that this is an official question? I had thought official questions are always correct.
Math Expert
Joined: 02 Sep 2009
Posts: 52231
Re: If zy < xy < 0, is |x-z| + |x| = |z|?  [#permalink]

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14 Mar 2017, 22:39
1
malavika1 wrote:
Bunuel wrote:
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.

Hi Bunuel, your explanation is totally convincing.

Can you or someone please confirm that this is an official question? I had thought official questions are always correct.

Yes, it's a question from old GMAT Prep. It's one of the very few examples of flawed questions from the official source.
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Re: If zy < xy < 0, is |x-z| + |x| = |z|? &nbs [#permalink] 14 Mar 2017, 22:39
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