If x 1 and m 1 , is x m

Question

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

(1) m > x + 1
(2) m > 1/(x-1)

Bunuel · Accepted Answer

Is x\sqrt{m}<\sqrt{(m+x+1)} --> square both sides (we can safely do this as both sides are positive) --> mx^2 mx^2-m m(x^2-1) m(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. Answer: B. Hope it's clear.

vinay.kaipra · Answer

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.

ParvezDhamani · Answer

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

Bunuel · Answer

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 .

ParvezDhamani · Answer

Thanks a lot for the explanation, but how do we take such hypothetical values in the real test in less than 2 mins, is there a easy way out?

Hussain15 · Answer

Perfect Solution Bunuel! +1 .

stonecold · Answer

B is sufficient as the question reduces to m(x-1)<1

stonecold · Answer

Also the first part of the statement can be proven insufficient by taking examples as x=1.0000000001 and m=100
and x=20 and m = 400
Remember => Do not discard statement 1 as it does not satisfies the inequality directly => rather look for values and prove it both sufficient and insufficient

lostincr · Answer

How can we square it we do not know the signs of x and m

what is x = -3 , m =9 then on plugging these values we will get -9 < sqrt(7) and on squaring the sign will change

Bunuel · Answer

Generally, we can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).

As for your example, notice that we are told that x > 1 and m > 1.

bumpbot · Answer

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If x > 1 and m > 1 , is x m < (m + x + 1) ?

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