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# If zy < xy < 0, is |x-z| + |x| = |z| a) z < x b) y

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If zy < xy < 0, is |x-z| + |x| = |z| a) z < x b) y [#permalink]  28 Jan 2010, 02:50
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If zy < xy < 0, is |x-z| + |x| = |z|

a) z < x
b) 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?
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Re: GMATprep Inequalities [#permalink]  28 Jan 2010, 03:51
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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

D is the answer i think
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Re: GMATprep Inequalities [#permalink]  28 Jan 2010, 06:14
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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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Re: GMATprep Inequalities [#permalink]  31 Jan 2010, 12:40
wow, I solved this question by myself:)
Re: GMATprep Inequalities   [#permalink] 31 Jan 2010, 12:40
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