If x is positive and x y, which of the following must be positive?

Question

If x is positive and x > y, which of the following must be positive?
I. x^2y - xy^2 II. x^3 - x^2y III. x^3 - xy^2 A. I only B. II only C. III only D. I and II E. II and III https://gmatclub.com/forum/download/file.php?id=82233 GMAT-Club-Forum-39l24jmk.png

Bunuel · Accepted Answer

Firstly, let's note that y could be positive as well as negative. Also, x > y implies that x - y > 0. Analyze each option:

I.  x^2y - xy^2

Factor out xy to get  xy(x - y) . As discussed, x - y must be positive, however xy could be positive for a positive y as well as negative for a negative y. Thus, this option is not necessarily positive.

II.  x^3 - x^2y

Factor out x^2 to get  x^2(x - y) . Both factors, x^2 and x - y, are positive, hence their product must be positive.

III.  x^3 - xy^2

Factor out x to get  x(x^2 - y^2) . The first factor, x, is positive. However, does x > y imply that x^2 > y^2? Or, to paraphrase, does x > y imply that x is farther from zero than y? No. For example, consider x = 1 and y = -10. Thus, this option is not necessarily positive.

Answer: B (II only ).

GMAT695 · Answer

Thank you @bunuel.

DanTheGMATMan · Answer

X is positive, y is either positive or negative, and x-y is positive so just factor out these answer choices and look at the possibilities:

https://www.youtube.com/watch?v=_KUZEANS4AQ

Su7sH · Answer

thanks Bunuel
I am unable to understand why doesnt the third option i.e., II.  x^3 - xy^2  , give negative outcome when I use the (x^2 - y^2) property to expand it to
x(x+y)(x-y).
I think it might be such that the use of the (x^2 - y^2) property in itself is incorrect in this case but I dont know why?

Bunuel · Answer

x^3 - xy^2 = x(x^2 - y^2) = x(x-y)(x+y)=(positive)(positive)(x + y)

If x + y is positive, then the result will be positive.  For example, x = 2 and y = 1.

If x + y is negative, then the result will be negative.  For example, x = 2 and y = -3.

Hope this helps.

Su7sH · Answer

Thanks Bunuel. 
Its fairly straightforward I dont know why I didnt see this. I think I should take a short break.

Posted from my mobile device

MBAToronto2024 · Answer

KarishmaB  - questions like these I'm solving by substitution of the real numbers. But here it failed. In a similar, question, you solved it by substitution. Can we solve it here as well?

KarishmaB · Answer

For a "must be true" kind of question, evaluating the expression is a far better approach as done by  Bunuel  above. If one were to try plugging in numbers, one would have to try various sets to ensure that it holds in every case. In a "could be true" question, plugging in numbers could work well. If you have the link of the other question, share with me. I will tell you why I did there what I did.

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If x is positive and x > y, which of the following must be positive?

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