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## Is x^2 > 1/x ?

tagged by: Brent@GMATPrepNow

This topic has 3 expert replies and 0 member replies

### Top Member

Vincen Legendary Member
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#### Is x^2 > 1/x ?

Thu Sep 14, 2017 1:48 pm
Is x^2 > 1/x ?

(1) x^2 > x

(2) 1 > 1/x

The OA is D.

Really each statement alone is sufficient? Why?

### GMAT/MBA Expert

Brent@GMATPrepNow GMAT Instructor
Joined
08 Dec 2008
Posted:
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Fri Sep 15, 2017 10:49 am
Vincen wrote:
Is xÂ² > 1/x ?

(1) xÂ² > x
(2) 1 > 1/x
Target question: Is xÂ² > 1/x ?

Statement 1: xÂ² > x
First off, this inequality tells us that x â‰  0
Second, we can conclude that xÂ² is POSITIVE.
So, we can safely divide both sides of the inequality by xÂ² to get: 1 > 1/x
If 1 > 1/x, then there are two possible cases:
Case a: x > 1. If x is a positive number greater than 1, then 1/x will definitely be less than 1.
Case b: x is negative. If x is negative, then 1/x will definitely be less than 1.

IMPORTANT: So how do these two cases affect the answer to the target question? Let's find out.
Case a: If x > 1, then xÂ² is greater than 1, AND 1/x is less than 1. This means xÂ² > 1/x
Case b: If x is negative, then xÂ² is positive, AND 1/x is negative. This means xÂ² > 1/x
Perfect - in both cases, we get the SAME answer to the target question
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 1 > 1/x
Notice that this inequality is the SAME as the inequality derived from statement 1 (we got 1 > 1/x)
Since we already saw that statement 1 is sufficient, it must be the case that statement 2 is also SUFFICIENT

Cheers,
Brent

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### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
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Fri Sep 15, 2017 9:43 am
Hi Vincen,

We're asked if X^2 is greater than 1/X. This is a YES/NO question. You can solve it by TESTing VALUES or using Number Properties.

1) X^2 > X

With the information in Fact 1, there are 2 'groups' of numbers that fit this inequality:
-ANY negative number
-Positive numbers that are GREATER than 1

IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
Thus, the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

2) 1 > 1/X

With the information in Fact 1, there are 2 'groups' of numbers that fit this inequality:
-ANY negative number
-Positive numbers that are GREATER than 1

These are the SAME 2 groups of numbers that fit Fact 1 - so we already know the answer to the question:
IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
Thus, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

GMAT assassins aren't born, they're made,
Rich

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### GMAT/MBA Expert

Jay@ManhattanReview GMAT Instructor
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Fri Sep 15, 2017 12:41 am
Vincen wrote:
Is x^2 > 1/x ?

(1) x^2 > x

(2) 1 > 1/x

The OA is D.

Really each statement alone is sufficient? Why?
Statement 1: x^2 > x

We can consider three important ranges.

1. If x is negative, then x^2 is always a positive number and x is always a negative number, thus, x^2 is greater than x.
2. If x > 1, then x^2 is always greater than x.

3. If 0 < x < 1, the inequality x^2 > x will not hold true. For example, say x = 1/2, then x^2 = 1/4 < 1/2 (= x). So, this is not a valid case.

From #1 and #2, we get either x < 0 or x > 1.

1. If x < 0, the ineuqality x^2 (= positive) > 1/x (= negative). The asnwer is Yes.
2. If x > 1, say x = 2, then x^2 (= 4) > x (= 1/2). The asnwer is Yes.

Statement 2: 1 > 1/x

Again, we can consider three important ranges.

1. If x is negative, then 1/x is always a negative number, thus, 1 is greater than 1/x.
2. If x > 1, then 1 is always greater than 1/x.

3. If 0 < x < 1, the inequality 1 > 1/x will not hold true. For example, say x = 1/2, then 1 < 1/(1/2) => 1 < 2. So, this is not a valid case.

From #1 and #2, we get either x < 0 or x >1.

This is the same result that we got from Statement 1. Thus, each statement itself is sufficient.

Hope this helps!

-Jay
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