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# Is 5^(x+2)/25<1 ?

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Is 5^(x+2)/25<1 ? [#permalink]  25 Jun 2012, 02:07
Expert's post
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Difficulty:

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Question Stats:

76% (01:48) correct 24% (00:49) wrong based on 379 sessions
Is $$\frac{5^{x+2}}{25}<1$$ ?

(1) $$5^x < 1$$
(2) $$x < 0$$

Diagnostic Test
Question: 33
Page: 25
Difficulty: 600
[Reveal] Spoiler: OA

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Re: Is 5^(x+2)/25<1 ? [#permalink]  25 Jun 2012, 02:07
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SOLUTION

Is $$\frac{5^{x+2}}{25}<1$$ ?

Is $$\frac{5^{x+2}}{25} <1$$? --> is $$\frac{5^{x+2}}{5^2}<1$$? --> is $$5^x<1$$? --> is $$x<0$$?

Or: is $$\frac{5^{x+2}}{25} <1$$? --> is $$5^{x+2}<5^2$$? --> is $$x+2<2$$? --> is $$x<0$$?

(1) $$5^x < 1$$ --> $$x<0$$. Sufficient.

(2) $$x < 0$$. Directly answers the question. Sufficient.

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Re: Is 5^(x+2)/25<1 ? [#permalink]  25 Jun 2012, 07:35
Hi,

Difficulty level: 600

To check, $$\frac {5^{(x+2)}}{25}<1$$
or $$(5^25^x)/25 < 1$$
or $$5^x < 1$$?

Using (1),
$$5^x<1$$. Sufficient.

Using (2),
x < 0
or $$5^x < 1$$. Sufficient.

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Re: Is 5^(x+2)/25<1 ? [#permalink]  29 Jun 2012, 03:43
Expert's post
SOLUTION

Is $$\frac{5^{x+2}}{25}<1$$ ?

Is $$\frac{5^{x+2}}{25} <1$$? --> is $$\frac{5^{x+2}}{5^2}<1$$? --> is $$5^x<1$$? --> is $$x<0$$?

Or: is $$\frac{5^{x+2}}{25} <1$$? --> is $$5^{x+2}<5^2$$? --> is $$x+2<2$$? --> is $$x<0$$?

(1) $$5^x < 1$$ --> $$x<0$$. Sufficient.

(2) $$x < 0$$. Directly answers the question. Sufficient.

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Re: Is 5^(x+2)/25<1 ? [#permalink]  06 Jul 2014, 03:10
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Is 5^(x+2)/25<1 ? [#permalink]  30 Jun 2015, 01:02
What would happen if we would have $$x=\frac{-1}{10}$$?

Am I correct that $$5^\frac{-1}{10} = \frac{1}{5^(1/10)} = \frac{1}{\sqrt[10]{5^1}}$$, but the denominator would still be always greater than the numerator?

Thank you!
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Is 5^(x+2)/25<1 ? [#permalink]  30 Jun 2015, 02:31
1
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bgpower wrote:
What would happen if we would have $$x=\frac{-1}{10}$$?

Am I correct that $$5^\frac{-1}{10} = \frac{1}{5^(1/10)} = \frac{1}{\sqrt[10]{5^1}}$$, but the denominator would still be always greater than the numerator?

Thank you!

Hello bgpower

You are correct that $$5^\frac{-1}{10} = \frac{1}{5^(1/10)} = \frac{1}{\sqrt[10]{5^1}}$$

But this statement "the denominator would still be always greater than the numerator?" is suspicious.
If you talk about this fraction: $$\frac{1}{\sqrt[10]{5^1}}$$ than you are wrong because in this fraction denominator $$\sqrt[10]{5^1}$$ is less than nominator $$1$$
If you talk about fraction from initial task then you are right denominator $$25$$ is bigger than nominator $$\sqrt[10]{5^1}$$
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Is 5^(x+2)/25<1 ? [#permalink]  30 Jun 2015, 02:42

I was actually talking about: $$\frac{1}{\sqrt[10]{5^1}}$$. So in this case I am wrong, as the numerator of 1 would clearly be greater than the denominator. But doesn't this mean that $$5^x$$ is not always <1. The question does not define x to be positive and as shown above if x is a negative fraction (as $$-\frac{1}{10}$$) then the result is greater than 1.

Thanks for the clarification!
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Re: Is 5^(x+2)/25<1 ? [#permalink]  30 Jun 2015, 03:39
1
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bgpower wrote:

I was actually talking about: $$\frac{1}{\sqrt[10]{5^1}}$$. So in this case I am wrong, as the numerator of 1 would clearly be greater than the denominator. But doesn't this mean that $$5^x$$ is not always <1. The question does not define x to be positive and as shown above if x is a negative fraction (as $$-\frac{1}{10}$$) then the result is greater than 1.

Thanks for the clarification!

I think you leave out last step of task (dividing on 25) and this confuse you.

You are absolutely right that $$\frac{1}{\sqrt[10]{5^1}}$$ greater than 1
but if we divide this result on 25 (as tasks asks) then result will be less than 1

----

This task can be solved in much faster way:

Tasks asks if $$\frac{5^{(x+2)}}{25}$$ will be less than 1

Let's transform this equation to $$5^{(x+2)} < 25$$ --> $$5^{(x+2)} < 5^2$$ from this view we see that this equation will be true if x will be less than 0

1) $$5^x<1$$ this is possible only if $$x < 0$$ - Sufficient
2) $$x<0$$ - this is exactly what we seek - Sufficient

Sometimes picking numbers is good but in this case algebraic way is much faster and at the end you will not have any hesitations in answer
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Is 5^(x+2)/25<1 ? [#permalink]  30 Jun 2015, 04:10
1
KUDOS
I actually did it the following way:

$$\frac{5^(x+2)}{25}<1$$
$$\frac{(5^x)(5^2)}{5^2}<1$$ => Here you can basically cancel out $$5^2$$ and are left with $$5^x<1$$

Here we come to the point we have already discussed.

(1) is clearly SUFFICIENT as it says exactly $$5^x<1$$.
(2) I thought (2) is NOT SUFFICIENT as for negative fractions (think $$-\frac{1}{10}$$), IMO this does NOT hold true, while for other negative values it does.
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Re: Is 5^(x+2)/25<1 ? [#permalink]  30 Jun 2015, 10:40
Expert's post
Is (5^x+2)/25<1 ?
Multiply both sides by 25 to yield 5^x+2>25. Rewrite 25 so that we have 5 as a base on both sides, so we want to know if 5^x+2> 5^2. The question is now is x+2>2. This will be true if x is negative.
(1) 5^x<1
1 can be rewritten so as 5^0 so as to give the same base. x<0 Sufficient.
(2) x<0
This is the same information as we derived from Statement 1 (x is negative). Therefore it is also sufficient.

D
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# Special Offer: Save up to $200 + GMAT Club Tests www.optimus-prep.com/gmat Optimus Prep Instructor Joined: 06 Nov 2014 Posts: 1006 Followers: 14 Kudos [?]: 168 [1] , given: 6 Re: Is 5^(x+2)/25<1 ? [#permalink] 30 Jun 2015, 10:41 1 This post received KUDOS Expert's post Is (5^x+2)/25<1 ? Multiply both sides by 25 to yield 5^x+2<25. Rewrite 25 so that we have 5 as a base on both sides, so we want to know if 5^x+2< 5^2. The question is now is x+2<2. This will be true if x is negative. (1) 5^x<1 1 can be rewritten so as 5^0 so as to give the same base. x<0 Sufficient. (2) x<0 This is the same information as we derived from Statement 1 (x is negative). Therefore it is also sufficient. D _________________ # Janielle Williams Customer Support # Special Offer: Save up to$200 + GMAT Club Tests

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Is 5^(x+2)/25<1 ? [#permalink]  01 Jul 2015, 02:56
OptimusPrepJanielle

Thank you for your explanation! I fully get your explanation. Nevertheless, I still don't see anyone addressing my questions, which basically asks what happens when x is a negative fraction as $$-\frac{1}{10}$$? I may be the one with an error here, but please explain it to me.

Thanks!
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Re: Is 5^(x+2)/25<1 ? [#permalink]  01 Jul 2015, 03:09
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Hi bgpower,

The issue is whether or not x is negative. If x is a negative fraction such as -1/10 the expression should still hold true. I hope that helps.
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Re: Is 5^(x+2)/25<1 ? [#permalink]  01 Jul 2015, 03:12
1
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bgpower wrote:
OptimusPrepJanielle

Thank you for your explanation! I fully get your explanation. Nevertheless, I still don't see anyone addressing my questions, which basically asks what happens when x is a negative fraction as $$-\frac{1}{10}$$? I may be the one with an error here, but please explain it to me.

Thanks!

Hello bgpower

When x = -1/10 then $$5^{-1/10+2} = 5^{19/10}$$
We need to check whether this $$5^{19/10}$$ less than $$5^2$$
19/10 less than 2
so $$5^{x+2} < 5^2$$
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Re: Is 5^(x+2)/25<1 ?   [#permalink] 01 Jul 2015, 03:12
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