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

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If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  17 Sep 2010, 11:10
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If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0
[Reveal] Spoiler: OA
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If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  17 Sep 2010, 14:58
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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 happens then the answer would 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.

Hope it helps.
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Re: Absolute values [#permalink]  17 Sep 2010, 11:33
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Hi, rgtiwari
Draw a number line and put 0 in the middle of it. now look at the data you are given. zy<xy<0.
we don't know if y is positive or negative. if y>0 than z is on the left of x which is on the left of zero => z<x and both z & x are originally negative.
the opposite if y<0, x will be on the right of zero and z will be on the right of x. => z>x. and both are originally positive. in both cases it seems that the equation is correct. but don't bother to check that just look at the extra data and continue from there.
If you find it hard to understand with variables just use numbers instead.
now let's look at (1) it tells you that z<x so we know that this is the first case only => the equality is definitely correct (just use numbers that maintain the data in the first case) - Sufficient.
(2) it tells you that y>0. again we are at the first case. - Sufficient.

pay attention that this is not a coincidence that the extra data given leads you in (1) and (2) to the same conclusion. if it doesn't you should suspect whether it's D or not.

Hope that helps.
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Re: If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  20 Jun 2012, 04:05
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rgtiwari wrote:
If zy < xy < 0, is | x - z | + |x| = |z|?

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

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).
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Re: If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  17 Sep 2012, 23:44
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rgtiwari wrote:
If zy < xy < 0, is | x - z | + |x| = |z|?

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

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.

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Re: If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  20 Dec 2012, 19:50
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eaakbari wrote:

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?

Before discussing this part, I have discussed in my post that absolute value of z must be greater than absolute value of x.
If absolute value of z is not greater than absolute value of x, then | z - x |= |z| - |x| does not hold when x and z have the same sign.

Since |z| must be greater than |x|,
| x - z | = |x| - |z| does not hold.
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Re: Absolute values [#permalink]  17 Sep 2010, 23:47
Got the approach. Thanks Rafi and Bunuel.
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Re: If zy<xy<0 is |x-z| + |x| = |z|? 1. z<x>0 2. [#permalink]  17 Sep 2012, 14:16
successstory wrote:
If zy<xy<0 is |x-z| + |x| = |z|?

1. z<x>0
2. y>0

why not a?

The question surprisingly does not require either statements for it to be true.

|x-z| + |x| = |z| can be rearranged as |x-z| = |z|-|x|. Now since we have an equal sign( opposed to an inequality, in which we would have to check both sides to make sure they are both positive or both negative to square (and flip)), we square both sides to get (|x-z|)^2 = (|z|-|x|)^2 which yields ===> xz=|x||z|

So our question becomes: Is xz=|x||z|?

Well this can only be true if (x>0 and z>0) OR (x<0 and z<0). In other words x and z must both be the same sign for the above statement to be true.

Now we are given zy<xy<0 as a fact. Two cases arise (+)(-)<(+)(-)<0 or (-)(+)<(-)(+)<0. Notice in both cases x and z are always the same sign!!!!! This statement is true from the gecko. Hence whatever unnecessary statement GMAC tells us will be sufficient. This is an usual problem you wont see again.
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Re: Absolute values [#permalink]  26 Nov 2012, 17:54
Is this really a 600 level question? Given its time consuming nature, seems more like a 750 level one.

Bunuel wrote:
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.

Hope it helps.
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Re: If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  20 Dec 2012, 08:32
VeritasPrepKarishma wrote:
Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign.

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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Re: If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  08 Feb 2013, 11:18
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.
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if zy<xy<0 is Ix-zI+IxI=IzI [#permalink]  15 May 2013, 05:30
from given

y(z-x) < 0 thus z is not = x and x is not equal to zero

if we square the 2 sides of the question

we get

x^2 + z^2 - 2xz +2x^2 - 2xz +x^2 = z^2 this boiles down to is 4x ( x-z) = 0 ? the answer is yes if x = 0 or x=z and no if we can know for sure that neother x=z nor x = 0

and this is given in the question stem.... neither givens are needed ..........as the answer is for sure NO
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Re: If zy < xy < 0, is | x - z | + |x| = |z|? [#permalink]  18 Jun 2014, 09:13
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Re: If zy < xy < 0, is | x - z | + |x| = |z|?   [#permalink] 18 Jun 2014, 09:13
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