If (x-y)^3 (x-y)^2, then which of the following must be true?

Question

If (x-y)^3>(x-y)^2 , then which of the following must be true? A. x^3y^2 D. x^5>y^4 E. x^3>y^3

kungfury42 · Answer

The Left hand side can be written as (x-y)²*(x-y) and now we can eliminate (x-y)² from both the sides because we know that (x-y)² is always a positive quantity (note that it can be 0 too but a quick glance at the options tells us there is an implicit assumption here that x≠y)

So now we are left with x-y>0 or x>y

Note that since our task here is to disprove the options, the best way to deal with such a situation is to take both the cases, both are negative numbers (x=-2 and y=-5) and both are positive numbers (x=5 and y=2) and use either case to disprove the given option.

(5)^3  = 125
 (2)^2  = 4
Clearly, 125 is not less than 4.  Eliminate option A.

(5)^5  = 3125
 (2)^4  = 16
Clearly, 3125 is not less than 16.  Eliminate option B.

(-2)^3  = -8
 (-5)^2  = 25
Clearly, -8 is not greater than 25.  Eliminate option C.

(-2)^5  = -32
 (-5)^4  = 625
Clearly, -32 is not greater than 625.  Eliminate option D.

(5)^3  = 125
 (2)^3  = 8
Clearly, 125 is greater than 8. Keep option E.

(-2)^3  = -8
 (-5)^3  = -125
Clearly, -8 is greater than -125. Keep option E.

We now know there will be no case where option E doesn't stand valid. Hence, our answer is option E.

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If (x-y)^3 > (x-y)^2, then which of the following must be true?

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