Brent@GMATPrepNow wrote:x, y and p are integers, and xyp ≠ 0. If p^x < p^y, which of the following MUST be true?

i) x − y < 0
ii) x < 2y
iii) x^p < y^p

A) i only
B) ii only
C) iii only
D) i and ii only
E) none of the above

Difficulty level: 650-700
Answer: E
Source: www.gmatprepnow.com

Two important rules:

ODD exponents preserve the sign of the base.
So, (NEGATIVE)^(ODD integer) = NEGATIVE
and (POSITIVE)^(ODD integer) = POSITIVE

An EVEN exponent always yields a positive result (unless the base = 0)
So, (NEGATIVE)^(EVEN integer) = POSITIVE
and (POSITIVE)^(EVEN integer) = POSITIVE
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So, one solution to the inequality p^x < p^y is p = -1, x = 7 and y = 2
Plugging those values into the inequality, we get: (-1)^7 < (-1)^2
Simplify to get: -1 < 1, WORKS.

Now plug p = -1, x = 7 and y = 2 into the three statements to get:

i) 7 - 2 < 0
Simplify to get: 5 < 0
NOT true.
So, statement i need not be true.

ii) 7 < 2(2)
Simplify to get: 7 < 4
NOT true.
So, statement ii need not be true.

iii) 7^(-1)} < 2^(-1)
Simplify to get: 1/7 < 1/2
This is TRUE.
So, we can't (yet) conclude that statement iii need not be true.

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Let's see if any other values will show that statement iii need not be true.

Another solution to the inequality p^x < p^y is p = -1, x = 1 and y = 2
Plugging those values into the inequality, we get: (-1)^1 < (-1)^2
Simplify to get: -1 < 1... WORKS.

Now plug p = -1, x = 1 and y = 2 into statement iii to get:
iii) 1^(-1) < 2^(-1)
Simplify to get: 1/1 < 1/2
NOT true.
So, statement iii need not be true.

Answer: E

Cheers,
Brent