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Re: Interesting Inequality [#permalink]
20 Jun 2010, 08:09

2

This post received KUDOS

Expert's post

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This post was BOOKMARKED

Hussain15 wrote:

If \(x > 1\) and \(m > 1\), is \(x\sqrt{m} < \sqrt{(m + x + 1)}\) ?

(1) \(m > x + 1\)

(2) \(m > \frac{1}{x-1}\)

Is \(x\sqrt{m}<\sqrt{(m+x+1)}\) --> square both sides (we can safely do this as both sides are positive) --> \(mx^2<m+x+1\) --> \(mx^2-m<x+1\) --> \(m(x^2-1)<x+1\) --> \(m(x-1)(x+1)<x+1\) --> reduce by \(x+1\) (we can safely do this as \(x+1>0\)) --> \(m(x-1)<1\).

So, finally the question becomes: is \(m(x-1)<1\)?

(1) \(m>x+1\) --> Not sufficient.

(2) \(m>\frac{1}{x-1}\) --> cross multiply (again we can safely doth is as \(x-1>0\)) --> \(m(x-1)>1\). Sufficient.

Re: Interesting Inequality [#permalink]
20 Jun 2010, 08:48

Bunuel, another feather in your hat. Good explanation. I was eagerly waiting for your answer as I was not able to solve the same Thanks once again. _________________

------------------------------------- Please give kudos, if my post is helpful.

Re: Interesting Inequality [#permalink]
21 Jun 2010, 02:30

Expert's post

ParvezDhamani wrote:

Can you explain why statement 1 alone doesnt suffice, it is given both x and m are >1

Because if \(m\) is some number, let's say \(m=100\) and \(x\) is very small (not violating the condition \(x>1\)), let's say \(x=1.000001\), then: \(x\sqrt{m}=10.00001<\sqrt{(m + x + 1)}\approx{10.01}\).

Bit if \(m=9\) and \(x=6\), then \(x\sqrt{m}=18>\sqrt{(m + x + 1)}=4\). _________________

Re: If x > 1 and m > 1 , is x m < (m + x + 1) ? [#permalink]
28 May 2015, 05:10

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

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