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# S99-06

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Math Expert
Joined: 02 Sep 2009
Posts: 43789

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16 Sep 2014, 00:53
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Difficulty:

85% (hard)

Question Stats:

38% (01:19) correct 62% (01:58) wrong based on 81 sessions

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Is $$(x - 2)^2 \gt x^2$$?

(1) $$x^2 \gt x$$

(2) $$\frac{1}{x} \gt 0$$
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Sep 2009
Posts: 43789

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16 Sep 2014, 00:53
Official Solution:

Rephrase the question by expanding the left side of the inequality:

Is $$x^2 - 4x + 4 \gt x^2$$?

Is $$-4x + 4 \gt 0$$?

Is $$4 \gt 4x$$?

Is $$1 \gt x$$?

Statement 1: INSUFFICIENT. The values of $$x$$ for which $$x^2 \gt x$$ are either negative or greater than 1. (Test numbers to prove this.) However, we do not know whether $$x$$ is less than or greater than 1.

Statement 2: INSUFFICIENT. The expression $$\frac{1}{x}$$ is positive when $$x$$ itself is positive. However, again we do not know whether $$x$$ is less than or greater than 1.

Statements 1 &amp; 2 together: SUFFICIENT. Combining the two conditions, we see that $$x$$ must be greater than 1. This provides a definitive "No" answer to the given question, and thus we have sufficiency.

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13 May 2017, 09:28
Looking at the second condition of 1/x >1. it is imperative that x<1 for all x>0, So, I thought condition 2 will be sufficient....Dunno whether I am right?
Math Expert
Joined: 02 Sep 2009
Posts: 43789

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13 May 2017, 09:54
amrit3982 wrote:
Looking at the second condition of 1/x >1. it is imperative that x<1 for all x>0, So, I thought condition 2 will be sufficient....Dunno whether I am right?

The question asks whether x < 1, while 1/x > 0 means that x > 0. So, the second statement is not sufficient: if x = 1/2, then the answer will be YES but if x = 2, then the answer will be NO.
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Re: S99-06   [#permalink] 13 May 2017, 09:54
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# S99-06

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