Jeff Miller

GMAT OFFICIAL GUIDE DS – If x is an integer…

If x is an integer…

Solution:

We are given that x is an integer, and we must determine whether x|x| < 2^x.

Statement One Alone: 

x< 0

Since we are given that x is less than 0, we can substitute negative integers for x into the inequality x|x| < 2^x.

x = -1

Is -1|-1| < 2^-1 ?

Is -1 * 1 < 2^-1 ?

Is -1 <½ ?

Yes!

Without substituting any other negative integers for x, we see that x|x| will always be negative when x is less than zero, and 2^x will always remain positive, because the base 2 is positive and a positive base raised to any power is always positive. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone: 

x = -10

Using the same logic that we used in statement one, we see that when x = -10, x|x| is less than 2^x. Statement two alone is sufficient to answer the question.

Answer: D

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