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(x^2 + x + 1)^x > 1

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Senior Manager
Joined: 21 Dec 2009
Posts: 453
Concentration: Entrepreneurship, Finance
(x^2 + x + 1)^x > 1  [#permalink]

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10 Jun 2010, 11:28
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Question Stats:

20% (01:38) correct 80% (01:39) wrong based on 30 sessions

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(x^2 + x + 1)^x > 1

1. (-infinity, 1)
2. (-1, infinity)
3. (0, infinity)
4. (1, infinity)
5. (0, infinity)
Math Expert
Joined: 02 Sep 2009
Posts: 60460
Re: (x^2 + x + 1)^x > 1  [#permalink]

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10 Jun 2010, 14:46
gmatbull wrote:
(x^2 + x + 1)^x > 1

1. (-infinity, 1)
2. (-1, infinity)
3. (0, infinity)
4. (1, infinity)
5. (0, infinity)

Not a GMAT question.

First let's check when $$(x^2 + x + 1)^x=1$$:
When the power is zero: - $$x=0$$;
or
When the base is 1: $$x^2 + x + 1=1$$ --> $$x=0$$ or $$x=-1$$.

So 2 values, 3 ranges to check (checking the values of $$x^2 + x + 1$$ for different values of $$x$$):
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$$x\leq{-1}$$ --> $$x^2 + x + 1\geq{1}$$ --> the number more than one in power less than -1 (remember power is $$x$$, which is $$\leq{-1}$$) will be less than one (check $$3^{(-2)}=\frac{1}{9}<1$$). So this range is not OK;

$$-1<x<0$$ --> $$\frac{3}{4}\leq{x^2 + x + 1}<1$$ --> the positive fraction less than one in negative power more than -1 will be more than 1 (check $$(\frac{3}{4})^{(-\frac{1}{2})}=\sqrt{{\frac{4}{3}}}>1$$). So this range is OK;

$$x>0$$ --> $$x^2 + x + 1>{1}$$ --> the number more than 1 in positive power will be more than 1. So this range is also OK.

So we got the ranges $$-1<x<0$$ and $$x>0$$ (remember when $$x=0$$ or $$x=-1$$, then $$(x^2 + x + 1)^x=1$$ so these values are not ok, we should exclude them).

No correct choice in answers, maybe it's meant to be 2, but the range (-1,infinity) includes 0 and it's not ok, also it's not clear whether -1 is included in the range or not.

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Senior Manager
Joined: 21 Dec 2009
Posts: 453
Concentration: Entrepreneurship, Finance
Re: (x^2 + x + 1)^x > 1  [#permalink]

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10 Jun 2010, 15:00
I always appreciate your quick response to my (sometimes daunting) questions.
Many thanks....
Math Expert
Joined: 02 Sep 2009
Posts: 60460
Re: (x^2 + x + 1)^x > 1  [#permalink]

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10 Jun 2010, 15:07
gmatbull wrote:
I always appreciate your quick response to my (sometimes daunting) questions.
Many thanks....

Can you pleas provide the source of this question (link maybe) and the OA.
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Senior Manager
Joined: 21 Dec 2009
Posts: 453
Concentration: Entrepreneurship, Finance
Re: (x^2 + x + 1)^x > 1  [#permalink]

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10 Jun 2010, 16:00
Bunuel wrote:
gmatbull wrote:
I always appreciate your quick response to my (sometimes daunting) questions.
Many thanks....

Can you pleas provide the source of this question (link maybe) and the OA.

I can recall the OA given (B); hint: use of log. Source:can't recall, but I will check and revert.
Thanks
Re: (x^2 + x + 1)^x > 1   [#permalink] 10 Jun 2010, 16:00
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