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Intern  Joined: 30 Sep 2010
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If x is negative, is x < -3 ?  [#permalink]

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Question Stats: 57% (01:30) correct 43% (01:31) wrong based on 2275 sessions

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If x is negative, is x < –3 ?

(1) x^2 > 9
(2) x^3 < –9

Originally posted by surendar26 on 02 Jan 2011, 08:47.
Last edited by ENGRTOMBA2018 on 12 Sep 2015, 09:44, edited 2 times in total.
Renamed the topic and edited the question.
Math Expert V
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Re: If x is negative, is x < -3 ?  [#permalink]

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23
If x is negative, is x < –3 ?

(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.

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Re: If x is negative, is x < -3 ?  [#permalink]

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5
2
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.

Ans:"A"
##### General Discussion
Manager  Joined: 25 Jul 2012
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Re: If x is negative, is x < -3 ?  [#permalink]

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I'm not seeing something in testing out statement 2.

Would someone be able to illustrate on a number line when testing out x=-3 and x=-4 only -64 <-3 and not -27?

And if you have a similar problem, please post.
Math Expert V
Joined: 02 Sep 2009
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Re: If x is negative, is x < -3 ?  [#permalink]

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DelSingh wrote:
I'm not seeing something in testing out statement 2.

Would someone be able to illustrate on a number line when testing out x=-3 and x=-4 only -64 <-3 and not -27?

And if you have a similar problem, please post.

Not sure I understand what you mean there. Can you please elaborate? Thank you.

Anyway both -64 and -27 are less than -3, since both are negative and further from 0 than -3 is.
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Re: If x is negative, is x < -3 ?  [#permalink]

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2
1
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.
Math Expert V
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Re: If x is negative, is x < -3 ?  [#permalink]

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3
DelSingh wrote:
Bunuel wrote:
DelSingh wrote:
I'm not seeing something in testing out statement 2.

Would someone be able to illustrate on a number line when testing out x=-3 and x=-4 only -64 <-3 and not -27?

And if you have a similar problem, please post.

Not sure I understand what you mean there. Can you please elaborate? Thank you.

Anyway both -64 and -27 are less than -3, since both are negative and further from 0 than -3 is.

Hopefully this will show what I am doing wrong in statement two: I had sufficient for statement 2 but that was wrong

x = -3 and x = -4 satisfy the second statement: (-3)^3 < -9 and (-4)^3 < -9.

Now, if x = -4 we have an YES answer because -4 < -3;
But if x = -3 we have a NO answer because -3 = -3 (x equals to -3 and is not less than -3).

Hope it's clear.
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Re: If x is negative, is x < -3 ?  [#permalink]

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1
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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If x is negative, is x < -3 ?  [#permalink]

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1
1
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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Re: If x is negative, is x < -3 ?  [#permalink]

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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...

Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: If x is negative, is x < -3 ?  [#permalink]

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3
JJSHHShank wrote:
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...

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:

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

DS Inequalities Problems
PS Inequalities Problems

700+ Inequalities problems

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

Hope this helps.
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Re: If x is negative, is x < -3 ?  [#permalink]

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JJSHHShank wrote:
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...

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.

Hope this helps.
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Re: If x is negative, is x < -3 ?  [#permalink]

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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...
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Re: If x is negative, is x < -3 ?  [#permalink]

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Bunnel,

Can you please explain why we should consider statement 1 as sufficient? I eliminated it thinking there two different values and not 1 value.

What am I missing here!

Shinrai
Math Expert V
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Re: If x is negative, is x < -3 ?  [#permalink]

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shinrai15 wrote:
Bunnel,

Can you please explain why we should consider statement 1 as sufficient? I eliminated it thinking there two different values and not 1 value.

What am I missing here!

Shinrai

If x is negative, is x < –3 ?

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

Please tell if anything is unclear above.
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If x is negative, is x < -3 ?  [#permalink]

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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
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GMAT 1: 610 Q48 V26 Re: If x is negative, is x < -3 ?  [#permalink]

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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
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If x is negative, is x < -3 ?  [#permalink]

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2
siddyj94 wrote:
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 $$\sqrt{-9}$$ but you can get that since (-2)^3 = -8, then $$\sqrt{-9}$$ 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.
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GMAT 1: 610 Q48 V26 Re: If x is negative, is x < -3 ?  [#permalink]

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Thanks Bunuel.
Doubt is cleared.
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Re: If x is negative, is x < -3 ?  [#permalink]

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surendar26 wrote:
If x is negative, is x < –3 ?

(1) x^2 > 9
(2) x^3 < –9

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.

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