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# Is 3^(x+2)/9 > 1 ?

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Is 3^(x+2)/9 > 1 ? [#permalink]  26 Jun 2012, 11:25
1
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Difficulty:

35% (medium)

Question Stats:

66% (01:58) correct 34% (01:10) wrong based on 147 sessions
Is 3^(x+2)/9 >1 ?

(1) 9^x > 1
(2) x > 0

I solved it. For example, in clue (1), I did this:
$$9^x > 1$$
$$9^x > 9^0$$
Then, $$x > 0$$, SUFFICIENT.

However, I would like to know whether there is a more conceptual approach. Specifically, whether in which cases x can be positive or negative.

Source: http://www.gmathacks.com
[Reveal] Spoiler: OA

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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  26 Jun 2012, 11:35
1
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Expert's post
Is 3^(x+2)/9 >1 ?

Is $$\frac{3^{x+2}}{9} >1$$? --> is $$\frac{3^{x+2}}{3^2} >1$$? --> is $$3^x>1$$? --> is $$x>0$$?

Or: is $$\frac{3^{x+2}}{9} >1$$? --> is $$3^{x+2}>3^2$$? --> is $$x+2>2$$? --> is $$x>0$$?

(1) 9^x > 1 --> $$x>0$$. Sufficient.
(2) x > 0. Directly answers the question. Sufficient.

Hope it's clear.
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  26 Jun 2012, 11:49
I am getting A as the answer.

Question is 3^x+2/9 = 3^x*9/9 = 3^x

From Option A
9^x>1 means 3^2x>1..
(3^x)2>1 . so 3^x has to be positive as it can never be negative for positive values or negative values of x.
so 3^x is always greater than 1
So we can answer this question.
But with statement B.. if x>0 then it can be 0.00001 also. so 3^0.0001 is less than 1 and with higher value like 2 we get value greater than 1. so not possible to answer with B. So answer is A

With
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  26 Jun 2012, 11:55
Expert's post
akalyan wrote:
I am getting A as the answer.

Question is 3^x+2/9 = 3^x*9/9 = 3^x

From Option A
9^x>1 means 3^2x>1..
(3^x)2>1 . so 3^x has to be positive as it can never be negative for positive values or negative values of x.
so 3^x is always greater than 1
So we can answer this question.
But with statement B.. if x>0 then it can be 0.00001 also. so 3^0.0001 is less than 1 and with higher value like 2 we get value greater than 1. so not possible to answer with B. So answer is A

With

The red part is not correct: $$3^{0.0001}\approx{1.000109867}>1$$.

Generally $$3^x>1$$ to hold true $$x$$ must be more than zero, hence $$3^x>1$$ simply means that $$x>0$$.

For more on number theory and exponents check: math-number-theory-88376.html

DS questions on exponents: search.php?search_id=tag&tag_id=39
PS questions on exponents: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope it helps.
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  26 Jun 2012, 14:53
akalyan wrote:
I am getting A as the answer.

Question is 3^x+2/9 = 3^x*9/9 = 3^x

From Option A
9^x>1 means 3^2x>1..
(3^x)2>1 . so 3^x has to be positive as it can never be negative for positive values or negative values of x.
so 3^x is always greater than 1
So we can answer this question.
But with statement B.. if x>0 then it can be 0.00001 also. so 3^0.0001 is less than 1 and with higher value like 2 we get value greater than 1. so not possible to answer with B. So answer is A

With

hey akalyan try taking log with base 3 to the question and you will easily reach at Is x>0 ?

Note everywhere in this answer wherever I write log it means log with base 3

Question is
3^(x+2)/9>1
Taking log both sides we get
log(3^(x+2)/9) > log 1
---> log(3^x+2) - log9>0
---->x+2 -2 >0
----> x>0
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  26 Jun 2012, 20:54
Thanks for correcting me bunuel.

Yes taking log is also good way to solve this problem manimani.
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  27 Jun 2012, 07:44
Hi Bunuel,

I would like to know whether you have a link with theory about exponents with inequalities.

For example, we cannot say inmediately that because $$a^x > a^y$$ , then $$x > y$$, right?

Because $$a$$ could be a fraction.

Thanks!
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  22 Oct 2013, 04:42
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Re: Is 3^(x+2)/9 > 1 ? [#permalink]  22 Oct 2013, 04:45
This was so cool you guys! I got it wrong and at first I didn't understand any of the explanations, but once I followed them and wrote them down, I got it!! Thanks! ....As you can probably guess, I'm new here. I promise not to be so annoying, once I get the hang of this....
Re: Is 3^(x+2)/9 > 1 ?   [#permalink] 22 Oct 2013, 04:45
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