If x is a nonzero number, which of the three numbers x, 1/x

If x is a nonzero number, which of the three numbers x, 1/x and x^2 is greatest?

(1) -1 < x < 1. If x=1/2, then x=1/2, 1/x=2, and x^2=1/4, so 1/x has the greatest value but if x is negative, then both x and 1/x are negative and x^2=positive, so x^2 has the greatest value. Not sufficient.

(2) x is negative. For negative x both x and 1/x are negative and x^2=positive, so x^2 has the greatest value. Sufficient.

Answer: B.

Hope it's clear.
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Hi sidney123,

I hope you have got your answer from the solution above. Just adding a few points:

1. Whenever you come across with such questions, look out for expressions which are independent of the sign of their base variables. In this question \(x^2\) is independent of the sign of \(x\). It will be always be non-negative irrespective of the sign of \(x\). Any even power of a variable will be independent of the sign of the variable whereas any odd power of a variable will be dependent on the sign of the variable

2. Do keep in mind certain ranges where numbers behave differently.

a. For example in the range \(0 < x < 1\), \(x^2 < x\) and \(x < \frac{1}{x}\).

b.Another such range is \(-1 < x < 0\) where \(x^2 > x\) but \(x^3 < x^2\) and \(x > \frac{1}{x}\)

Hope this helps

Regards
Harsh
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If x is a nonzero number, which of the three numbers x, 1/x and x^2 is greatest?

(1) -1 < x < 1 We know that x is a fraction, but we don't know if it is negative or positive. If x is -1/2, then we have -1/2, -2, and 1/4 (x^2 is greatest). If x is 1/2 we have 1/2, 2, and 1/4, making (1/x greatest)
(2) x is negative We can answer this knowing the sign of x (again, take -1/2 as the example)
B

Hi sidney123,

Your approach to TEST VALUES is a good one, but you have to do the necessary work (on the pad, so you can "see" it) to make sure that you're selecting the correct answer.

Here, we're asked to consider X, 1/X and X^2.....We're asked which is GREATEST.

Fact 2 tells us that X is NEGATIVE....

Using your two examples, we'd end up with....

X = -2
1/X = -1/2
X^2 = (-2)^2 = 4
So X^2 is the greatest

X = -1/2
1/X = 1/(-1/2) = -2
X^2 = (-1/2)^2 = 1/4
So X^2 is the greatest

The two examples yield the SAME result. Based on what we're told in Fact 2, X will ALWAYS be NEGATIVE, 1/X will ALWAYS be NEGATIVE and X^2 will ALWAYS be POSITIVE, so the answer will ALWAYS be X^2. Thus, Fact 2 is SUFFICIENT.

As you continue to practice DS, whenever you think a Fact is insufficient, you should be able to quickly prove it. Doing that little bit of extra work can pick you up a LOT of points in the Quant section.

Final Answer:

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

Thankyou for all your input guys, really appreciate the time and effort taken to explain the simple problem. For some reason, I read the Q wrong(I don't know why?), I was looking at arranging x, 1/x and x^2 from smallest to greatest, in which case we need both statements. But the Q asks for which is greatest and stmt.2 is sufficient.

Given: x is a nonzero number

Target question: Which of the three numbers x, 1/x and x² is greatest?

Statement 1: -1 < x < 1
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 1/2. This means 1/x = 1/(1/2) = 2, and x² = (1/2)² = 1/4. In this case, the answer to the target question is 1/x is the greatest value
Case b: x = -1/2. This means 1/x = 1/(-1/2) = -2, and x² = (-1/2)² = 1/4. In this case, the answer to the target question is x² is the greatest value
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of testing values when a statement doesn't feel sufficient, read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: x is negative
This means 1/x = 1/negative = negative, and x² = (negative,)² = positive.
Since x² is positive, and x and 1/x are negative, the answer to the target question is x² is definitely the greatest value
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

(1) Clearly insufficient.

(2) If \(x < 0\), then lets say \(x = -1\)

\(x = -1\)
\(\frac{1}{x} = -1\)
\(x^2 = 1\)

\(x^2\) is the greatest. SUFFICIENT.

Answer is B.
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