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# If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo

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If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  10 Nov 2012, 13:40
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Question Stats:

72% (02:50) correct 28% (02:34) wrong based on 280 sessions
If $$x < y < z$$ but $$x^2 > y^2 > z^2 > 0$$, which of the following must be positive?

A.$$x^3$$ $$y^4 z^5$$

B. $$x^3 y^5 z^4$$

C. $$x^4 y^3 z^5$$

D. $$x^4 y^5 z^3$$

E. $$x^5 y^4 z^3$$
[Reveal] Spoiler: OA

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Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  10 Nov 2012, 17:28
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carcass wrote:
If $$x < y < z$$ but $$x^2 > y^2 > z^2 > 0$$, which of the following must be positive?

A.$$x^3$$ $$y^4 z^5$$

B. $$x^3 y^5 z^4$$

C. $$x^4 y^3 z^5$$

D. $$x^4 y^5 z^3$$

E. $$x^5 y^4 z^3$$

Think of the cases in which '$$x < y < z$$ but $$x^2 > y^2 > z^2 > 0$$' happens.

A simple case I can think of is all negative numbers: -5 < -4 < -3 but 25 > 16 > 9 > 0
Another thing that comes to mind is that z can be positive as long as its absolute value remains low: -5 < -4 < 3 but 25 > 16 > 9 > 0

We need to find the option that must stay positive:

A.$$x^3$$ $$y^4 z^5$$
Will be negative in this case: -5 < -4 < 3
i.e. x negative, y negative, z positive

B. $$x^3 y^5 z^4$$
Will be positive in both the cases.

C. $$x^4 y^3 z^5$$
Will be negative in this case: -5 < -4 < 3
i.e. x negative, y negative, z positive

D. $$x^4 y^5 z^3$$
Will be negative in this case: -5 < -4 < 3
i.e. x negative, y negative, z positive

E. $$x^5 y^4 z^3$$
Will be negative in this case: -5 < -4 < 3
i.e. x negative, y negative, z positive

Notice that for an expression to stay positive, we need the power of both x and y to be either even or both to be odd since x and y are both negative. Also, we need the power of z to be even so that it doesn't affect the sign of the expression. Only (B) satisfies these conditions.
We don't need to consider any other numbers since we have already rejected 4 options using these numbers. The fifth must be positive in all cases.
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Kudos [?]: 41050 [2] , given: 5661

Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  11 Nov 2012, 04:07
2
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Expert's post
1
This post was
BOOKMARKED
carcass wrote:
If $$x < y < z$$ but $$x^2 > y^2 > z^2 > 0$$, which of the following must be positive?

A.$$x^3$$ $$y^4 z^5$$

B. $$x^3 y^5 z^4$$

C. $$x^4 y^3 z^5$$

D. $$x^4 y^5 z^3$$

E. $$x^5 y^4 z^3$$

First of all: $$x^2 > y^2 > z^2 > 0$$ means that $$|x|>|y|>|z|>0$$ (we can take even roots from all parts of an inequality, if all parts are non-negative).

Thus we have that $$x < y < z$$ and $$|x|>|y|>|z|>0$$. This implies that both $$x$$ and $$y$$ must be negative numbers: $$x$$ to be less than $$y$$ and at the same time to be further from zero than $$y$$ is, it must be negative. The same way $$y$$ to be less than $$z$$ and at the same time to be further from zero than $$z$$ is, it must be negative. Notice here, that $$z$$ may be positive as well as negative. For example if $$x=-3$$, $$y=-2$$, then $$z$$ can be -1 as well as 1. Since we don't know the sign of $$z$$, then in order to ensure (to guarantee) that the product will be positive its power in the expression must be even. Only answer choice B fits.

Hope it's clear.
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Kudos [?]: 41050 [0], given: 5661

Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  11 Nov 2012, 04:09
Expert's post
carcass wrote:
If $$x < y < z$$ but $$x^2 > y^2 > z^2 > 0$$, which of the following must be positive?

A.$$x^3$$ $$y^4 z^5$$

B. $$x^3 y^5 z^4$$

C. $$x^4 y^3 z^5$$

D. $$x^4 y^5 z^3$$

E. $$x^5 y^4 z^3$$

Similar question, also from MGMAT, to practice: if-a-b-c-and-d-are-integers-and-ab2c3d4-0-which-of-the-136450.html

Hope it helps.
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Kudos [?]: 3376 [0], given: 720

Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  11 Nov 2012, 09:53
Expert's post
This is one of that question to bookmark

Thanks
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Kudos [?]: 41050 [0], given: 5661

Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  17 Jun 2013, 04:51
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

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Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  21 Aug 2013, 04:37
Bunuel wrote:
carcass wrote:
If $$x < y < z$$ but $$x^2 > y^2 > z^2 > 0$$, which of the following must be positive?

A.$$x^3$$ $$y^4 z^5$$

B. $$x^3 y^5 z^4$$

C. $$x^4 y^3 z^5$$

D. $$x^4 y^5 z^3$$

E. $$x^5 y^4 z^3$$

First of all: $$x^2 > y^2 > z^2 > 0$$ means that $$|x|>|y|>|z|>0$$ (we can take even roots from all part of an inequality, if all parts are non-negative).

Thus we have that $$x < y < z$$ and $$|x|>|y|>|z|>0$$. This implies that both $$x$$ and $$y$$ must be negative numbers: $$x$$ to be less than $$y$$ and at the same time to be further from zero than $$y$$ is, it must be negative. The same way $$y$$ to be less than $$z$$ and at the same time to be further from zero than $$z$$ is, it must be negative. Notice here, that $$z$$ may be positive as well as negative. For example if $$x=-3$$, $$y=-2$$, then $$z$$ can be -1 as well as 1. Since we don't know the sign of $$z$$, then in order to ensure (top guarantee) that the product will be positive its power in the expression must be even. Only answer choice B fits.

Hope it's clear.

Thanx Bunuel for the explanation..it's now crystal clear..
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Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  28 Mar 2014, 09:25
Play Smart, try to look for similarities across answer choices

Z doesn't necessarily have to be negative. See we have that x and y have their directions reversed when we square them, hence they are negative strictly speaking/ But x could be positive because we are not told that z<0 in the first equation. Therefore only B works

Cheers
J
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If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo [#permalink]  17 Jul 2014, 04:51
$$x^ay^bz^c>0$$
There are two possible ways:
1) x<y<z<0
a+b+c must be even
2) x<y<0<z
a+b must be even, and B is the only answer
If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo   [#permalink] 17 Jul 2014, 04:51
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