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OG Is x|x| < 2^x?

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Source: — Data Sufficiency |

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by [email protected] » Wed Aug 23, 2017 9:46 am
Hi AbeNeedsAnswers,

We're told that X is an integer. We're asked if X|X| < 2^X. This is a YES/NO question. We can answer it with a bit of Number Property knowledge.

1) X < 0

With Fact 1, we know that X is NEGATIVE. By definition, that means...
X|X| = (Neg)|Neg| = Negative
2^(Negative) = Positive
Thus, X|X| will ALWAYS be less than 2^X and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

2) X = -10

With the value of X, we can absolutely answer the question (we would just need to plug in that value:
Is (-10)|-10| < 2^(-10)?
The answer to the question IS yes, but we don't have to actually do that work. There would be just one answer to the question, so it doesn't really matter what that one answer is.
Fact 2 is SUFFICIENT

Final Answer: D

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Rich
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by Brent@GMATPrepNow » Wed Aug 23, 2017 12:50 pm
AbeNeedsAnswers wrote:If x is an integer, is x|x|<2^x ?

(1) x < 0
(2) x = -10

D
Target question: Is x|x|< 2^x ?

Given: x is an integer

Statement 1: x < 0
In other words, x is NEGATIVE
So, x|x| = (NEGATIVE)(|NEGATIVE|) = (NEGATIVE)(POSITIVE) = NEGATIVE

IMPORTANT: 2^x will be POSITIVE for all values of x.

Since x|x| must be NEGATIVE, and since 2^x must be POSITIVE, we can be certain that x|x|< 2^x
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = -10
So, x|x| = (-10)(|-10|) = (-10)(10) = -100 = a NEGATIVE
On the other hand, 2^x = 2^(-10) = 1/(2^10) = some POSITIVE number
Since x|x| is NEGATIVE, and since 2^x must be POSITIVE, we can be certain that x|x|< 2^x
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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