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\[ Is x^{12} - 2x^{11} negative? \]

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\[ Is x^{12} - 2x^{11} negative? \]

by Gmat_mission » Sun May 06, 2018 10:44 am

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$$Is\ \ \ x^{12}-2x^{11}\ \ \ negative\ \ ?$$
$$(1)\ \ \ x^2<|x|$$ $$(2)\ \ \ x^{-1}<-1$$ [spoiler]OA=B[/spoiler].

Can someone explain this question to me? I'd be thankful. How can I determine which is the correct answer?
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Source: — Data Sufficiency |
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by Vincen » Sun May 06, 2018 12:04 pm

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Hello Gmat_mission.

Let's take a look at the question.

First, let's rewrite the given expression as follows: $$x^{12}-2x^{11}=x^{11}\left(x-2\right)=x^{10}\cdot x\cdot\left(x-2\right).$$

(1) If x^2 < |x| then -1 < x < 1. Now:

1. If x is positive, then x-2<0, and hence the given expression is negative. The answer, in this case, is YES.
2. If x is negative, then x-2<0, and hence the given expression is positive. The answer, in this case, is NO.

Therefore, this statement is NOT SUFFICIENT.

(2) If x^(-1) <-1 then this implies that -1 < x < 0.

Now, x-2 < 0 and hence the given expression is positive. The answer, in this case, is NO.

Since we get only one answer, this statement is SUFFICIENT.

Hence, the answer is the option C.

I hope it helps.
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