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Re: Is x > y^2? 1. x > y + 5 2. x^2 - y^2 = 0 [#permalink]
22 Jun 2012, 06:35

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Expert's post

Is x>y^2?

(1) x>y+5 --> x-y>5. Clearly insufficient, for example: if x=1 and y=-10 then the answer is NO, but if x=10 and y=1 then the answer is YES. Two different answers, hence not sufficient.

(2) x^2-y^2=0 --> (x-y)(x+y)=0 --> so either x-y=0 or x+y=0. Also insufficient: if x=1 and y=1, then answer is NO, buy if x=\frac{1}{2} and y=\frac{1}{2}, then the answer is YES. Two different answers, hence not sufficient.

(1)+(2) As from (1) x-y>5\neq{0}, then from (2) must be true that x+y=0 --> so x=-y --> substitute x in (1) --> -y-y>5 --> y<-\frac{5}{2}<0, as x=-y, then x>\frac{5}{2}>0, so y^2 (or which is the same x^2) will always be more than x, thus the answer to the question "Is x>y^2" is NO. Sufficient.

To elaborate more as x=-y>0, the only chance for x>y^2 to hold true (or which is the same for x>x^2 to hold true) would be if x is fraction (0<x<1). For example if x=\frac{1}{2} and y=-\frac{1}{2} then x=\frac{1}{2}>y^2=\frac{1}{4}. But the fact that x>\frac{5}{2}>0 rules out this option.