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
------------------------------------
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.
-------------------------------------
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
