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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

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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Hi guys...this is how i solved the question... plz correct me,wher i am wrong... For statement 1: x^2>9, x^2-9>0 , (x+3)(x-3)>0 , {x>-3 & x>3} or {x<-3 & x<3} Now considering only negative values i get x>-3 or x<-3.. Since answer is not consistent A is not sufficient...

Hi guys...this is how i solved the question... plz correct me,wher i am wrong... For statement 1: x^2>9, x^2-9>0 , (x+3)(x-3)>0 , {x>-3 & x>3} or {x<-3 & x<3} Now considering only negative values i get x>-3 or x<-3.. Since answer is not consistent A is not sufficient...

Thanks in advance

This is not correct.

x^2 > 9

|x| > 3 (by taking the square root from both sides, notice that we can safely do that because both sides are non-negative);

Hi guys...this is how i solved the question... plz correct me,wher i am wrong... For statement 1: x^2>9, x^2-9>0 , (x+3)(x-3)>0 , {x>-3 & x>3} or {x<-3 & x<3} Now considering only negative values i get x>-3 or x<-3.. Since answer is not consistent A is not sufficient...

Thanks in advance

Hey Mate,

If x^2 > 9 x > 3 or x < -3

With what you're saying is x > -3 then x = -2 for example and x^2 = 4 which is not in line with the equation, and this is also incorrect.

In terms of solving an inequality equation,

If x^2 > 9 then x > 3 or x < -3 If x^2 < 9 then -3<x<3

Try to input value of x in the range, and it'll always be true.

Thanks Bunuel and akshayk for the quick response... The links given were bible for inequalities...Thanks a lot. Understood that we cannot just solve taking a pen and a scratch pad... we need to think too-at every step, especially when dealing with inequalities...

There are no doubts about 1 as it is obviously sufficient . I want to discuss more about 2. 2. Is x<-3 if x=-3 then x^3=-27, -27<-9 which is very true . This is the very point where the option 2 fails. Is X<-3 ? No because 2 becomes true at x=-3 as-27<-9, so x cannot be <-3.If the question was is x<= -3 then it would have been sufficient. Ans - A Hope this helps Kudos if you like the solution

I know this is a lame question but i am trying to get into my head this topic. For the second statement can we pick values other than -3 and -4 like -2 and -5 and prove that the statement is not sufficient

I know this is a lame question but i am trying to get into my head this topic. For the second statement can we pick values other than -3 and -4 like -2 and -5 and prove that the statement is not sufficient

Numbers you pick to get that a statement is NOT sufficient, should a. satisfy that statement and b. should give a NO and an YES answer to the question.

x = -2 does not satisfy x^3 < -9, because x^3 in this case is -8, which is greater than -9, not less than it.

(2) x^3 < –9, implies that \(x < (\approx 2.1)\) (of course you are not expected to know what is \(\sqrt[3]{-9}\) but you can get that since (-2)^3 = -8, then \(\sqrt[3]{-9}\) will be LESS than -2 but greater than -3 (-3^3 = -27)). So any number from -2.1 to -3, inclusive, will give a No answer to the question and any number less than -3 will give an YES answer to the question.

We are given that x is negative, and we must determine whether x < -3.

Statement One Alone:

x^2 > 9

Taking the square root of both sides of the inequality in statement one we have:

√x^2 > √9

|x| > 3

x > 3 OR -x > 3

x > 3 OR x < -3

Since we are given that x is negative, we see that x must be less than -3. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

x^3 < -9

Using the information in statement two, we see that x can be less than -3 or not be less than -3.

For example, if x = -4, (-4)^3 = -64, (which fulfills the statement) and -4 is less than -3.

However, if x = -3, (-3)^3 = -27, (which fulfills the statement) but -3 is not less than -3.

Statement two alone is not sufficient to answer the question.

The answer is A.
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