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(1) x^2 > 9 --> \(x<-3\) or \(x>3\) as given that \(x<0\) then we have that \(x<-3\). Sufficient.

(2) x^3 < –9 --> if \(x=-3\) (\(x^3=-27<-9\)) then the answer will be NO (as \(x\) equals to -3 and is not less than -3) but if \(x=-4\) (\(x^3=-64<-9\)) then the answer will be YES. Not sufficient.

1. \(x^2>9\) \(|x|>3\) \(x>3 \hspace{3} or \hspace{3} x<-3\) We know that x is -ve. Thus; \(x<-3\) Sufficient.

2. \(x^3<-9\) \(x^3\) can be -27 making x=-3 or \(x^3\) can be -64 making x=-4[/m] We can't conclude that x is definitely smaller than -3. Not Sufficient.

they are asking is x<-3, not if x^3<-3. -27 and -64 are the values of x^3. so x=-3 and x=-4. negative 3 isn't less than negative 3, so answer is no. negative 4 is less than negative 3, so answer is yes. insufficient.

Statement 2 : X^3<-9 => We know (-2)^3 = -8 and (-3)^3= -27 => it isnt given in the q that x has to be an integer => x can be any decimal slightly less than -2.0 ie. -2.5^3 (-15) and thus give an answer NO & x can be any number <-3 (=>x^3 <-27)and give an answer YES. Thus, insufficient

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St1) Quite obviously SUFF St2) x^3<-9. First of all x has to be negative as cube root of a negative number will be negative. Now lets take cube root of both sides: x<- (~2.1) ...[we know than cube root of -8 is -2, so cube root of of -9 will be just slightly smaller than -2] So our ballpark estimate is that x lies to the left of -2.1 we don't know if it will lie to the left of -3. INSUF

Alternatively-

Stem: If x is negative, is x < -3 ? In other words, i) is x^2>9? (Inequality sign will flip) ii) is x^3<-27? (Ineqality sign doesnt change) ii) is x^4> 81? (Inequality sign will flip) etc etc . . . st1) Straight away yes from i)...SUF St2) only tells us x^3 is less than -9 so x^3 could be less than -27 or not. INSUF

Ans: A _________________

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