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Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1

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Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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New post Updated on: 14 May 2018, 01:24
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Is \(x^{12}- 2x^{11}\) negative?


(1) \(x^2 < |x|\)

(2) \(x^{-1} < -1\)

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Originally posted by reto on 10 Sep 2015, 02:47.
Last edited by Bunuel on 14 May 2018, 01:24, edited 3 times in total.
Renamed the topic and edited the question.
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Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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New post 10 Sep 2015, 03:13
1
reto wrote:
Is x^12 – 2x^11 negative?

(1) x^2 < |x|
(2) x^(-1) < -1


Simplifying,

x^11(x-2)<0--->?
This can only be possible when 0<x<2

1. x^2<|x|
-1<x<1

Not Sufficient.

2. 1/x<-1
-1<x<0
Answer is "No"

Sufficient.

Answer: B
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Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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New post 10 Sep 2015, 03:33
1
reto wrote:
Is x^12 – 2x^11 negative?

(1) x^2 < |x|
(2) x^(-1) < -1


Solution: x^12 – 2x^11 ==> x^11(x-2) It basically asks about x.
If x<=0 NO.
If 0<x<2 yes.
If x>=2 NO.

(1) Both LHS and RHS are non-negetive. x cannot be 0 and x^2 < x only when 0<x<1 or -1<x<0
Therefore, Yes or No. Insufficient.

(2)-x>1 ==> x<-1 ==> x^11<-1 and x-2<-1.
x^11(x-2) > 0 . NO. Sufficient

Option, B
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Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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New post 09 Mar 2016, 09:53
1
reto wrote:
Is \(x^{12}\)– \(2x^{11}\)negative?

(1) \(x^2\) < |x|
(2) \(x^{-1}\) < -1


The question asks whether , \(x^{12} – 2*x^{11}<0\) ---> \(x^{11}*(x-2)<0\). Remember that \(x^{odd} <0\) for \(x<0\)and \(x^{odd}>0\) for \(x>0\)

Per statement 1, \(x^2< |x|\) ---> carefully note this statement tells us that a squared positive quantity (x^2) < another positive quantity (|x|). This is only possible when -1<x<0 or 0<x<1. For these 2 ranges, you will get both "no" for -1<x<0 and "yes" for 0<x<1, making this statement NOT sufficient.

Per statement 2, \(x^{-1} < -1\) ---> (DO NOT MULTIPLY BY x as you do not know the sign of x) \(x^{-1} < -1 = x^{-1}+1 < 0\) ---> (x+1)/x < 0 ---> as the result of this fraction <0 ---> (x+1) and x are of OPPOSITE SIGNS.

2 cases :

Case 1: x+1 > 0 and x < 0 ---> x>-1 and x<0 ---> -1<x<0 and

Case 2: x+1<0 --> x<-1 and x>0 ---> not possible.

Thus, the only range possible -->-1<x<0 and for this range you get a NO for the question asked. Thus statement 2 is sufficient. B is thus the correct answer.

Hope this helps.
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Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1 &nbs [#permalink] 14 May 2018, 01:23
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