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# If n is not equal to 0, is |n| < 4 ?

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If n is not equal to 0, is |n| < 4 ? [#permalink]

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08 Aug 2009, 06:55
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If n is not equal to 0, is |n| < 4 ?

(1) n^2 > 16

(2) 1/|n| > n

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-n-is-not-equal-to-0-is-n-4-1-n-85256.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 17 Jul 2014, 01:30, edited 1 time in total.
Edited the question and added the OA.

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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08 Aug 2009, 07:24
lbsgmat wrote:
If n is not equal to 0, is |n| < 4 ?

(1) n2 > 16

(2) 1/|n| > n

D for me.
1) if n^2 > 16, then we get two possibilities, either n>4 or n<-4. For both cases, absolute value of n will be always > 4.
Sufficient.

2) In order for this inequality to be true, n can be anything < 1 (excluding 0 ).
So we cant say if the absolute value is < 4.
Insufficient.

Last edited by Economist on 08 Aug 2009, 10:02, edited 1 time in total.

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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08 Aug 2009, 07:30
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I would go for A

statement 1 is sufficient as n^2 > 16 meaning n > 4 or n < -4, either way, |n| > 4

statement 2 1/|n| > n...insufficient, two examples:
when n = -2, 1/2 > -2, |-2| < 4

when n = -5, 1/5 > -5, |-5| > 4

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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16 Aug 2009, 00:47
8) If n is not equal to 0, is |n| < 4 ?

(1) n^2 > 16

(2) 1/|n| > n

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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16 Aug 2009, 03:07
Imo D
given n!= 0, is |n| < 4? which is n^2 < 16?
stmt 1
n^2>16 , so n^2 cannot be < 16.
suffi.

stmt2
1/|n| > n
squaring on both sides
1/(n^2) > n^2
1 > n^4 ( can multiply because n^2 is always +ve independent of the value of n)
n^4<1, then defintely n^2 < 1, so n^2 < 16
suffi.

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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16 Aug 2009, 04:59
Yep D.

stmt 1:
n>4 or n<-4. In either case the magnitude of n (absolute value) will always be >4.
Sufficient.

stmt 2:
1>|n|*n, For this inequality to be true, n<1. Hence, |n| will always be <4.
Sufficient.

Last edited by Economist on 16 Aug 2009, 08:05, edited 1 time in total.

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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16 Aug 2009, 07:59
Economist wrote:
stmt 2:
1>|n|*n, For this inequality to be true, we have -1<n<0 or 0<n<1. Hence, |n| will always be <4.
Sufficient.

I agree with D, but the solution for statement 2 is not $$-1<n<0$$ or $$0<n<1$$

$$-1<n<0$$ or $$0<n<1$$ is the solution for $$n^2 < 1$$
Solution for $$|n|*n < 1$$ is $$n < 1$$
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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Aug 2009, 06:49
yezz wrote:
crejoc wrote:
Imo D
given n!= 0, is |n| < 4? which is n^2 < 16?
stmt 1
n^2>16 , so n^2 cannot be < 16.
suffi.

stmt2
1/|n| > n
squaring on both sides ( can we do that " what if n is -ve , squaring will hide the sign??)
1/(n^2) > n^2
1 > n^4 ( can multiply because n^2 is always +ve independent of the value of n)
n^4<1, then defintely n^2 < 1, so n^2 < 16
suffi.

We cannot do that unless we are sure that n is +ve, else the inequality sign will reverse. Hence we can only derive 1>|n|n, as |n| is always +ve:)

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Aug 2009, 11:34

n*|n| < 1 ==> n < 0 it holds good for any negative values so insuff to say if |n| < 4 or not...

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Aug 2009, 17:03
yezz wrote:
8) If n is not equal to 0, is |n| < 4 ?

(1) n^2 > 16

(2) 1/|n| > n

Thats A.

(1) If n^2 > 16, n is either > 4 or < -4. Suff..
(2) If 1/|n| > n, n is smaller than 1 but not 0 i.e. n could be 0.5, -1, -10 etc. not suff..

I was scrolling down for A. Got it............
skpMatcha wrote:

n*|n| < 1 ==> n < 0 it holds good for any negative values so insuff to say if |n| < 4 or not...

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Aug 2009, 17:05
Economist wrote:
squaring on both sides ( can we do that " what if n is -ve , squaring will hide the sign??)
We cannot do that unless we are sure that n is +ve, else the inequality sign will reverse. Hence we can only derive 1>|n|n, as |n| is always +ve:)

that makes sense economist,.. i thought squaring will hide it.. you are correct it cannot be squared, unless n is always +ve..

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Aug 2009, 17:29
I dont understand, doesn't statement 2 yield N to be under 1 and statement 1 yields it to be above 4. Can this be a question, N cant be both? Either way, arent both statement sufficient, statement 1 tells you it has to be over 4, statement 2 tells you definitively that its under it?

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Aug 2009, 18:04
sfeiner wrote:
I dont understand, doesn't statement 2 yield N to be under 1 and statement 1 yields it to be above 4. Can this be a question, N cant be both? Either way, arent both statement sufficient, statement 1 tells you it has to be over 4, statement 2 tells you definitively that its under it?

i think stmt1 is clear and is sufficient, and you have no doubt with it.

consider stmt2

question is |n|< 4?

stmt2 given
1/|n|>n

case 1:
take when n is +ve
1 > |n|* n
1> n^2
n^2<1
so -1<n<1
in this interval for any values |n| is < 4

case 2:
when n is -ve
1/|n|> n
1/(-n)>n
multiply by -n
1< -n^2
-n^2>1
n^2>-1
since n is negative, and n^2>-1, for all negative values of n this is true, so n can take any negative values say -1,-2..and so on

if n=-2
1/|n|>n, 1/2> -2 is true, check |n| < 4, |-2|<4, 2< 4 true,

if n= -8
1/|n|>n, 1/8 > -8 , but when you check |n|<4, |-8|<4? , no 8 is not < 4, so insufficient

Another easy way is to directly check by plugging numbers

stmt2
1/|n|> n

for any positive value of n, this statement holds false, so n can take negative values and only fractions.

check with negative numbers
when n = -2
|n| < 4? , |-2|<4 , yes
but when n = -8
|n| < 4?, |-8|is not < 4,
so stmt2 insufficient
similarly, for fractions also try using n = 1/2, n=1/8 , it is insufficient
hope this helps..

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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15 Sep 2009, 02:16
Agree with A. It is the sole choice, which represents a clear answer on the question.
In stmt B there is an evidence, that n is negative only. This gives nothing.
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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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23 Dec 2009, 17:21
Can somebody breakdown for me how n^2 > 16 is n<-4 or n>4.

As I look at it if n^2 > 16 then n > + or -4.

also I think I understand how |n| > 4 but please breakdown that as well.

Thanks.

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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23 Dec 2009, 18:52
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sudimba wrote:
Can somebody breakdown for me how n^2 > 16 is n<-4 or n>4.

As I look at it if n^2 > 16 then n > + or -4.

also I think I understand how |n| > 4 but please breakdown that as well.

Thanks.

n^2>16---->+-n>16---->+n>16 or -n>16---->n<-4(multiplying bth side by -1 reverses the sign)
one can try with numbers also as square of(-3)=9<16 therfore n<-4 to satisfy the inequality

same with otherone
lnl>4--->+n>4 or -n > 4---->n<-4
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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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24 Dec 2009, 16:38
From (i)

$$n^2>16$$
$$\sqrt{n^2}>\sqrt{16}$$
$$|n|>\sqrt{16}$$
$$|n|>4$$

/* Because $$|n| = \sqrt{n^2}$$ applies only for positive square roots of n */
Therefore,
|n|<4 is not TRUE.

From(i) we could find this answer.

From(ii)

$$1/|n|>|n|$$

$$n|n|<1$$
$$|n|<1/n$$
If n>0,
$$n<1/n$$
$$n^2<1$$
$$|n|<1$$
|n|<4 is TRUE.

or If n<0,
$$-n<1/n$$
$$]n>1/n$$
$$n^2>1$$
$$|n|>1$$
|n|<4 may or may not be TRUE.

Therefore from(ii), we can't answer this question.

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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Re: If n is not equal to 0, is |n| < 4 ? [#permalink]

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17 Jul 2014, 01:30
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If n is not equal to 0, is |n| < 4 ?

Question basically asks is -4<n<4 true.

(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.

(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.

As you can see we don't really want the complete range for (2) to see that this statement is not sufficient, but still if interested:

1/|n| > n --> n*|n| < 1.

If n<0, then we'll have -n^2<1 --> n^2>-1. Which is true. So, n*|n| < 1 holds true for any negative value of n.
If n>0, then we'll have n^2<1 --> -1<n<1. So, n*|n| < 1 also holds true for 0<n<1.

Thus 1/|n| > n holds true if n<0 and 0<n<1.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-n-is-not-equal-to-0-is-n-4-1-n-85256.html
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Re: If n is not equal to 0, is |n| < 4 ?   [#permalink] 17 Jul 2014, 01:30
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