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

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Math Expert
Joined: 02 Sep 2009
Posts: 60460
Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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09 Dec 2019, 00:56
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Difficulty:

75% (hard)

Question Stats:

57% (02:22) correct 43% (02:42) wrong based on 46 sessions

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Is $$x^{12}- 2x^{11}$$ negative?

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

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

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

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09 Dec 2019, 03:17
given
$$x^{12}- 2x^{11}$$ negative
#1
$$x^2 < |x|$$
x will be a fraction only possibility

but for x = + fraction we get no and x = -ve we get yes
insufficient
#2
$$x^{-1} < -1$$
is -ve ; possiblity ; x=-2,-3; we get NO
IMO B sufficient

Bunuel wrote:
Is $$x^{12}- 2x^{11}$$ negative?

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

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

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Senior Manager
Joined: 13 Feb 2018
Posts: 496
GMAT 1: 640 Q48 V28
Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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09 Dec 2019, 09:47
1
We can factor out $$x^{11}$$ to get
$$x^{11}*(x-2)$$

the only possibility for this expression to be negative is
when 0<x<2

1 stm --> gives us a range -1<x<1. Not sufficient
2 stm -- $$\frac{1}{x}<-1$$ means x is negative. Sufficient

IMO
Ans: B
Manager
Joined: 15 Jan 2018
Posts: 66
Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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09 Dec 2019, 10:23
LevanKhukhunashvili wrote:
We can factor out $$x^{11}$$ to get
$$x^{11}*(x-2)$$

the only possibility for this expression to be negative is
when 0<x<2

1 stm --> gives us a range -1<x<1. Not sufficient
2 stm -- $$\frac{1}{x}<-1$$ means x is negative. Sufficient

IMO
Ans: B

Actually, the first piece of information provided tells us that the range of x is between 0 and 1. When fractions are squared, they always get closer to 0, and for negative fractions, they will change signs and get closer to zero. So x must be positive for that to hold true. I could be wrong, but I believe that the answer is actually D.
Senior Manager
Joined: 13 Feb 2018
Posts: 496
GMAT 1: 640 Q48 V28
Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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09 Dec 2019, 10:36
1
edlc313 wrote:
LevanKhukhunashvili wrote:
We can factor out $$x^{11}$$ to get
$$x^{11}*(x-2)$$

the only possibility for this expression to be negative is
when 0<x<2

1 stm --> gives us a range -1<x<1. Not sufficient
2 stm -- $$\frac{1}{x}<-1$$ means x is negative. Sufficient

IMO
Ans: B

Actually, the first piece of information provided tells us that the range of x is between 0 and 1. When fractions are squared, they always get closer to 0, and for negative fractions, they will change signs and get closer to zero. So x must be positive for that to hold true. I could be wrong, but I believe that the answer is actually D.

IMO your reasoning about the first statement is not correct. if x=-1/2 that statement holds true, so the range is not 0<x<1
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Joined: 24 Nov 2016
Posts: 1068
Location: United States
Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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13 Dec 2019, 09:20
Quote:
Is $$x^{12}- 2x^{11}$$ negative?

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

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

$$x^{12}- 2x^{11}<0…x^{11}(x- 2)<0$$
$$x<0:x-2>0…x>2=false…(x<0)$$
$$x>0:x-2<0…x<2…0<x<2$$

We need to find if $$0<x<2$$

(1) $$x^2 < |x|$$ insufic

$$x^2 < |x|:-1<x<0…or…0<x<1$$

(2) $$x^{-1} < -1$$ sufic

$$x^{-1} < -1…1/x<-1…1/x+1<0…(1+x)/x<0$$
$$x>0:1+x<0…x<-1=false…(x>0)$$
$$x<0:1+x>0…x>-1…-1<x<0$$

Ans (B)
Intern
Joined: 08 Oct 2019
Posts: 6
Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1  [#permalink]

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16 Dec 2019, 02:19
[quote="Bunuel"]Is $$x^{12}- 2x^{11}$$ negative?

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

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

$$x^{12}- 2x^{11} < 0$$ <=> $$x^{11}*(x - 2) < 0$$ => check whether 0 < x < 2 or not

(1) -> -1 < x < 1. Insufficient as:
If -1 < x < 0: $$x^{11}*(x - 2) > 0$$
If 0 < x < 1: $$x^{11}*(x - 2) < 0$$

(2) -> -1 < x < 0 => $$x^{11}*(x - 2) > 0$$ => Sufficient

==> B
Re: Is x^12 – 2x^11 negative? (1) x^2 < |x| (2) x^(-1) < -1   [#permalink] 16 Dec 2019, 02:19
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