If x is negative, is x < –3 ? (1) x^2 > 9 (2) x^3 < –9

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

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);

x < -3 or x > 3.

You should brush up fundamentals on inequalities:

Inequalities Made Easy!

Solving Quadratic Inequalities - Graphic Approach
Inequality tips
Wavy Line Method Application - Complex Algebraic Inequalities

DS Inequalities Problems
PS Inequalities Problems

700+ Inequalities problems

https://gmatclub.com/forum/inequalities-trick-91482.html
https://gmatclub.com/forum/data-suff-ine ... 09078.html
https://gmatclub.com/forum/range-for-var ... 09468.html
https://gmatclub.com/forum/everything-is ... 08884.html
https://gmatclub.com/forum/graphic-appro ... 68037.html

Hope this helps.

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.

Hope it helps.

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.

Solution:

St (1):- x^2 > 9

=> x<−3 or x>3 (Using wavy curve approach)

Given x<0

=> Then x<−3 as x>3 can be eliminated. Sufficient.

St (2):- x^3 < –9

Let’s plug in values to check.

At x=−3

=> x^3=−27<−9

Q-stem – Is x <-3

Answer-NO (as x = -3 and is not less than -3) but

If x=−4

=> (x3=−64<−9) then the answer to the question stem will be YES.

Contradictory answers (Yes/No) –Insufficient

Hence option (a)

Devmitra Sen
GMAT SME

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Answer: Option A

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To determine whether x is less than −3 when x is negative, we need to consider the given statements:

(1) x^2>9


(2) x^3<−9


Let's analyze each statement individually:

Statement (1) states that x^2>9. Taking the square root of both sides, we have ∣x∣>3. This means that the absolute value of x is greater than 3. Since we are specifically considering the case where x is negative, if ∣x∣>3 holds true, then it implies that x is indeed less than −3. Therefore, statement (1) alone is sufficient to determine that x<−3.

Statement (2) states that x^3<−9. In this case, since x is negative, raising a negative number to an odd power results in another negative number. Therefore, x^3 will be negative. However, this statement does not provide enough information to determine whether x is less than −3 specifically. It only tells us that x^3 is negative but does not give any bounds or restrictions on the value of x. Therefore, statement (2) alone is insufficient to determine whether x<−3.

Since statement (1) alone is sufficient to determine that x<−3, the answer is (A) Statement (1) alone is sufficient, while statement (2) alone is not sufficient.