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# If N is a negative, which of the following must be true?

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If N is a negative, which of the following must be true? [#permalink]  13 Feb 2014, 07:40
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74% (01:46) correct 26% (00:35) wrong based on 77 sessions
If N is a negative, which of the following must be true?

I. $$N^3<N^2$$
II. $$N+\frac{1}{N}<0$$
III. $$N=\sqrt{N^2}$$

A. I only
B. II only
C. III only
D. I and III only
E. I and II only
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 Feb 2014, 09:31, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Integer Properties [#permalink]  13 Feb 2014, 09:28
Anasarah wrote:
I was going through a manual and found the question below. I thought the answer is A but it is not....

If N is a negative, which of the following must be true?

I. N^3<N^2 II. N+1/N<0 III.N=[square_root]N^2

a. I only b.II only c.III only d.I and III only e. I and II only

You figured out I and III - so will only comment on II.

N is negative. 1/N is negative. Adding any two negative numbers will be negative. So N + 1/N < 0 must be true.
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Re: If N is a negative, which of the following must be true? [#permalink]  13 Feb 2014, 09:37
Expert's post
Anasarah wrote:
If N is a negative, which of the following must be true?

I. $$N^3<N^2$$
II. $$N+\frac{1}{N}<0$$
III. $$N=\sqrt{N^2}$$

A. I only
B. II only
C. III only
D. I and III only
E. I and II only

If N is a negative, which of the following must be true?

I. $$N^3<N^2$$
II. $$N+\frac{1}{N}<0$$
III. $$N=\sqrt{N^2}$$

A. I only
B. II only
C. III only
D. I and III only
E. I and II only

I. $$N^3<N^2$$. Since N is negative, then (N^3=negative) < (N^2=positive). Hence, this one must be true,

II. $$N+\frac{1}{N}<0$$. Both N and 1/N are negative, the sum of two negative values is negative. The same here: must be true.

III. $$N=\sqrt{N^2}$$. The square root function cannot give negative result: $$\sqrt{some \ expression}\geq{0}$$. Negative N cannot equal to positive $$\sqrt{N^2}$$. Never true.

Hope it's clear.

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Re: If N is a negative, which of the following must be true? [#permalink]  13 Feb 2014, 14:32
Expert's post
Bunuel, from an algebraic standpoint, if we manipulate Statement II like below, why does the inequality leave open the possibility that N^2 can be a negative fraction? I get why Neg + Neg < 0, but was wondering about the below. Thank you.

N+(1/N)<0 -->
N<-(1/N)
N^2>-1
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Re: If N is a negative, which of the following must be true? [#permalink]  14 Feb 2014, 00:48
Expert's post
m3equals333 wrote:
Bunuel, from an algebraic standpoint, if we manipulate Statement II like below, why does the inequality leave open the possibility that N^2 can be a negative fraction? I get why Neg + Neg < 0, but was wondering about the below. Thank you.

N+(1/N)<0 -->
N<-(1/N)
N^2>-1

Not following you... We are asked to find which of the options must be true while given that N is negative (negative integer, negative fraction, negative irrational number). For negative N, N +1/N < 0 must be true. Can you please elaborate what you mean? Thank you.
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Re: If N is a negative, which of the following must be true? [#permalink]  14 Feb 2014, 15:38
Expert's post
Sry, my question was more of a general one. Assuming n is neg, I was playing around with the inequality to see if I could manipulate it to coincide with what was already quite apparent (negative + negative = negative).

Basically, I subtracted the negative fraction to the opposite side of the inequality and then multiplied the denominator to the original side (flipping the inequality in the process with N neg). I ended up with n^2 which is is presumed to be positive. Everything seemingly checks out as the inequality says n^2 is > -1, however this includes >=0 n^2 >-1 as well, which seems erroneous.

I was just wondering how to interpret this and if I am making any missteps in my algebraic manipulations and/or thought process.

Thanks very much for your help/insight.
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Re: If N is a negative, which of the following must be true? [#permalink]  17 Feb 2014, 06:57
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Expert's post
m3equals333 wrote:
Sry, my question was more of a general one. Assuming n is neg, I was playing around with the inequality to see if I could manipulate it to coincide with what was already quite apparent (negative + negative = negative).

Basically, I subtracted the negative fraction to the opposite side of the inequality and then multiplied the denominator to the original side (flipping the inequality in the process with N neg). I ended up with n^2 which is is presumed to be positive. Everything seemingly checks out as the inequality says n^2 is > -1, however this includes >=0 n^2 >-1 as well, which seems erroneous.

I was just wondering how to interpret this and if I am making any missteps in my algebraic manipulations and/or thought process.

Thanks very much for your help/insight.

I guess you want to solve for which range of n, n+1/n<0 holds true...

$$n+\frac{1}{n}<0$$ --> $$\frac{n^2+1}{n}=\frac{positive}{n}<0$$ --> positive/n to be negative, n must be negative, thus $$n+\frac{1}{n}<0$$ holds true for $$n<0$$.

Hope it helps.
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Re: If N is a negative, which of the following must be true? [#permalink]  17 Feb 2014, 07:57
Option E.
The first statement:N^3<N^2 is true because N=-ve =>N^3=-ve
And N^2=+ve since square is always +ve

The second statement is also true because N+1/N=(N^2+1)/N=-ve since N^2+1 will be +ve and N is given -ve.

No need to look at Statement 3 since no option says all three correct.

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Re: If N is a negative, which of the following must be true? [#permalink]  19 Feb 2014, 17:28
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Yes it does, much appreciated
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Re: If N is a negative, which of the following must be true?   [#permalink] 19 Feb 2014, 17:28
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