If x z^2 0, which of the follo

Question

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

KarishmaB · Accepted Answer

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.

Bunuel · Answer

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. Answer: B. Hope it's clear.

Bunuel · Answer

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.

Carcass · Answer

This is one of that question to bookmark

Thanks

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Theory on Inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
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Reetabrata Ghosh · Answer

Thanx Bunuel for the explanation..it's now crystal clear..

jlgdr · Answer

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

cht1128 · Answer

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

guialain · Answer

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.

laddaboy · Answer

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.

Rickooreoisb · Answer

When x,y 2>1 2>-1 -2<1 -2<-1 but in all cases x^2 > y^2 So from the question when it says x

Usernamevisible · Answer

to add when x^2>y^2 then x>y or xy^2 ; x has to be negative otherwise it wont be greater than y^2 so x is negative similarly from y and z we get y is negative from option 3 we get x negative, y negative and z we dont know the sign but z^4 will be positive so outcome positive

luisdicampo · Answer

Deconstructing the Question From x^2>y^2>z^2>0 , we get |x|>|y|>|z|>0 . Together with x

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

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