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Board of Directors D
Joined: 01 Sep 2010
Posts: 3417
If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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7
40 00:00

Difficulty:   45% (medium)

Question Stats: 69% (02:29) correct 31% (02:37) wrong based on 549 sessions

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

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Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9850
Location: Pune, India
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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8
8
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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Karishma
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##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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6
4
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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Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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

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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Board of Directors D
Joined: 01 Sep 2010
Posts: 3417
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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This is one of that question to bookmark Thanks
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Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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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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Intern  Joined: 21 May 2013
Posts: 22
Location: India
Concentration: Finance, Marketing
GMAT 1: 660 Q49 V32 Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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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.. SVP  Joined: 06 Sep 2013
Posts: 1545
Concentration: Finance
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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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
Intern  Joined: 02 Jul 2014
Posts: 10
Concentration: Marketing, Strategy
If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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2
$$x^ay^bz^c>0$$
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
Current Student S
Joined: 01 Dec 2016
Posts: 102
Concentration: Finance, Entrepreneurship
GMAT 1: 650 Q47 V34 WE: Investment Banking (Investment Banking)
If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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1
A trick to solve this question in 20 secondes.
Because you are looking for answer choice that must be positive (assuming the question stem), let's remove positve variables in each of the answer choices.

For example, in Option A - The sign of (x^3)*(y^4)*(z^5) is the same as sign of x*z
Using similar reasoning, you can rework the answer choices are follows:
we are looking for

A- x*z
B- x*y
C- y*z
D- y*z
E- x*z

Notice that: A and E are the same, and C and D are the same.

Therefore only B can be correct.
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What was previously considered impossible is now obvious reality.
In the past, people used to open doors with their hands. Today, doors open "by magic" when people approach them

Originally posted by guialain on 21 Apr 2017, 10:00.
Last edited by guialain on 22 Nov 2017, 13:37, edited 1 time in total.
Intern  B
Joined: 23 Aug 2017
Posts: 23
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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Hi Bunuel. What does it happen if x and y are positive no intenger numbers?

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 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.
Manager  G
Joined: 22 May 2015
Posts: 125
Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo  [#permalink]

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I plugged in values

first i considered all 3 as negative x,y,z = -3,-2,-1 and found all the options are positive

Next set of values -3,-2,1 and found only Option B is positive.
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Consistency is the Key Re: If x < y < z but x^2 > y^2 > z^2 > 0, which of the follo   [#permalink] 07 Jul 2019, 22:30
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