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# Which of the following represents the complete range of x

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Director
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Which of the following represents the complete range of x  [#permalink]

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08 Feb 2011, 09:29
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55% (02:05) correct 45% (01:47) wrong based on 2968 sessions

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Which of the following represents the complete range of x over which $$x^3 – 4x^5 < 0$$?

A. $$0 < |x| < \frac{1}{2}$$

B. $$|x| >\frac{1}{2}$$

C. $$–\frac{1}{2} < x < 0$$ or $$\frac{1}{2} < x$$

D. $$x < –\frac{1}{2}$$ or $$0 < x < \frac{1}{2}$$

E. $$x < –\frac{1}{2}$$ or $$x > 0$$

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Which of the following represents the complete range of x  [#permalink]

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08 Feb 2011, 09:41
32
25
gmatpapa wrote:
Which of the following represents the complete range of x over which x^3 - 4x^5 < 0?

(A) 0 < |x| < ½
(B) |x| > ½
(C) –½ < x < 0 or ½ < x
(D) x < –½ or 0 < x < ½
(E) x < –½ or x > 0

Basically we are asked to find the range of $$x$$ for which $$x^3-4x^5<0$$ is true.

$$x^3-4x^5<0$$;

$$x^3(1-4x^2)<0$$;

$$(1+2x)*x^3*(1-2x)<0$$:

"Roots" are -1/2, 0, and 1/2: $$-\frac{1}{2}<x<0$$ or $$x>\frac{1}{2}$$.

Check this for more: http://gmatclub.com/forum/inequalities-trick-91482.html
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Re: Which of the following represents the complete range of x  [#permalink]

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01 Nov 2012, 16:06
5
Bunuel
Thanx a trillion for your post on solving inequalities using graph
You know i paid over 300$to test prep institutes but got nothing out of it.......when i asked such basic question the tutor got frustrated and insulted me.....But hats off to you... MAx wat will i give 1 kudo...... Wat an expeirence it has been with GMAt club Thanx a lot Bunuel Trillion kudos to you and Hats off to you for addressing problems with patience..............I cant express myself how satisfied i am feeling. ##### General Discussion Senior Manager Joined: 08 Nov 2010 Posts: 351 WE 1: Business Development Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 09 Feb 2011, 23:41 Thanks Bunuel. +1 A question - what is the best way u use to know if the "good" area is above or below? i mean - what was the best way for u to know that its between -1/2 to 0 i used numbers ex. 1/4 but it consumes time! is there any better technique? thanks. _________________ Math Expert Joined: 02 Sep 2009 Posts: 47983 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 10 Feb 2011, 03:28 17 24 144144 wrote: Thanks Bunuel. +1 A question - what is the best way u use to know if the "good" area is above or below? i mean - what was the best way for u to know that its between -1/2 to 0 i used numbers ex. 1/4 but it consumes time! is there any better technique? thanks. Check the link in my previous post. There are beautiful explanations by gurpreetsingh and Karishma. General idea is as follows: We have: $$(1+2x)*x^3*(1-2x)<0$$ --> roots are -1/2, 0, and 1/2 (equate the expressions to zero to get the roots and list them in ascending order), this gives us 4 ranges: $$x<-\frac{1}{2}$$, $$-\frac{1}{2}<x<0$$, $$0<x<\frac{1}{2}$$ and $$x>\frac{1}{2}$$ --> now, test some extreme value: for example if $$x$$ is very large number than first two terms ((1+2x) and x) will be positive but the third term will be negative which gives the negative product, so when $$x>\frac{1}{2}$$ the expression is negative. Now the trick: as in the 4th range expression is negative then in 3rd it'll be positive, in 2nd it'l be negative again and finally in 1st it'll be positive: + - + -. So, the ranges when the expression is negative are: $$-\frac{1}{2}<x<0$$ (2nd range) or $$x>\frac{1}{2}$$ (4th range). Hope its clear. _________________ Retired Moderator Joined: 16 Nov 2010 Posts: 1458 Location: United States (IN) Concentration: Strategy, Technology Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 13 Feb 2011, 21:31 Hi Bunuel I'm getting D as answer : x^3(1-2x)(1+2x) < 0 $$-ve --- -1/2---- +ve--- 0----- -ve-----1/2--- +ve$$ Could you please explain where I'm wrong ? Regards, Subhash _________________ Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant) GMAT Club Premium Membership - big benefits and savings Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8195 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 13 Feb 2011, 22:05 24 14 subhashghosh wrote: Hi Bunuel I'm getting D as answer : x^3(1-2x)(1+2x) < 0 $$-ve --- -1/2---- +ve--- 0----- -ve-----1/2--- +ve$$ Could you please explain where I'm wrong ? Regards, Subhash Even though your question is directed to Bunuel, I will give a quick explanation. The concept of the rightmost section being positive is applicable when every term is positive in the rightmost region. This is the case whenever the expressions involved are of the form (x - a) or (ax - b) etc. When you have a term such as (1-2x), the rightmost region becomes negative. So either, as Bunuel mentioned, check for an extreme value of x or convert (1-2x) to (2x - 1) and flip the sign to >. _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Joined: 02 Sep 2009
Posts: 47983
Re: Which of the following represents the complete range of x  [#permalink]

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02 Mar 2011, 18:04
ajit257 wrote:
Bunuel wrote:
gmatpapa wrote:
Which of the following represents the complete range of x over which x^3 - 4x^5 < 0?

(A) 0 < |x| < ½
(B) |x| > ½
(C) –½ < x < 0 or ½ < x
(D) x < –½ or 0 < x < ½
(E) x < –½ or x > 0

Basically we are asked to find the range of $$x$$ for which $$x^3-4x^5<0$$ is true.

$$x^3-4x^5<0$$ --> $$x^3(1-4x^2)<0$$ --> $$(1+2x)*x^3*(1-2x)<0$$ --> roots are -1/2, 0, and 1/2 --> $$-\frac{1}{2}<x<0$$ or $$x>\frac{1}{2}$$.

Check this for more: inequalities-trick-91482.html

Bunuel...I got x<0, X>1/2 and x< -1/2. How do you get -1/2< x

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.
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Re: Which of the following represents the complete range of x  [#permalink]

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02 Mar 2011, 18:40
1
Karishma
I flipped the sign before. So I got x^3(2x-1)(2x-1) > 0

2 cases - both +ve or both -ve

case 1
-------
x > 0 and |x| > 1/2. Hence x > 1/2

case 2
------
x < 0 and 4x^2 - 1 < 0
x < 0 and -1/2 < x < 1/2
Taking the most restrictive value-
-1/2 < x < 0

I hope this is correct. Btw this is 750 level in 2 mins.

VeritasPrepKarishma wrote:
subhashghosh wrote:
Hi Bunuel

I'm getting D as answer :

x^3(1-2x)(1+2x) < 0

$$-ve --- -1/2---- +ve--- 0----- -ve-----1/2--- +ve$$
Could you please explain where I'm wrong ?

Regards,
Subhash

Even though your question is directed to Bunuel, I will give a quick explanation.

The concept of the rightmost section being positive is applicable when every term is positive in the rightmost region. This is the case whenever the expressions involved are of the form (x - a) or (ax - b) etc. When you have a term such as (1-2x), the rightmost region becomes negative. So either, as Bunuel mentioned, check for an extreme value of x or convert (1-2x) to (2x - 1) and flip the sign to >.
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Re: Which of the following represents the complete range of x  [#permalink]

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02 Mar 2011, 19:12
2
1
gmat1220 wrote:
Karishma
I flipped the sign before. So I got x^3(2x-1)(2x-1) > 0

2 cases - both +ve or both -ve

case 1
-------
x > 0 and |x| > 1/2. Hence x > 1/2

case 2
------
x < 0 and 4x^2 - 1 < 0
x < 0 and -1/2 < x < 1/2
Taking the most restrictive value-
-1/2 < x < 0

I hope this is correct. Btw this is 750 level in 2 mins.

Yes, it is correct... and since you know what you are doing, you will need to work very hard to fall short of time on GMAT.
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Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! Math Expert Joined: 02 Sep 2009 Posts: 47983 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 04 Mar 2011, 02:52 ajit257 wrote: Bunuel...I still did not get it. so i get |x| > 1/2 which gives me x>1/2 and x<-1/2 and x < 0. Please could you tell me where I am going wrong. Thanks for you patience. |x| > 1/2 means that x<-1/2 or x>1/2. The range you wrote is wrong also because x<-1/2 and x < 0 doesn't makes any sense. Check Walker's post on absolute values for more: math-absolute-value-modulus-86462.html _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8195 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 20 Jun 2012, 22:42 5 4 gmatpapa wrote: Which of the following represents the complete range of x over which x^3 - 4x^5 < 0? (A) 0 < |x| < ½ (B) |x| > ½ (C) –½ < x < 0 or ½ < x (D) x < –½ or 0 < x < ½ (E) x < –½ or x > 0 Responding to a pm: The problem is the same here. How do you solve this inequality: $$(1+2x)*x^3*(1-2x)<0$$ Again, there are 2 ways - The long algebraic method: When is $$(1+2x)*x^3*(1-2x)$$ negative? When only one of the terms is negative or all 3 are negative. There will be too many cases to consider so this is painful. The number line method: Multiply both sides of $$(1+2x)*x^3*(1-2x)<0$$ by -1 to get $$(2x + 1)*x^3*(2x - 1)>0$$ Take out 2 common to get $$2(x + 1/2)*x^3*2(x - 1/2)>0$$ [because you want each term to be of the form (x + a) or (x - a)] Now plot them on the number line and get the regions where this inequality holds. Basically, you need to go through this entire post: inequalities-trick-91482.html _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Joined: 27 Nov 2010
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Re: Which of the following represents the complete range of x  [#permalink]

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15 Nov 2012, 13:58
Hi All,

Could I conclude that for this case i.e (1+2x)*x^3*(1-2x)<0
even if one of the terms <0, that does not necessarily mean that the entire product of the 3 terms <0.
Cause like if the eq was (1+2x)*x^3*(1-2x)= 0 ....I could have safely concluded that
However in this case for the entire product <0.. either 1 terms or 2 terms or even all 3 terms can be - ve.
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Re: Which of the following represents the complete range of x  [#permalink]

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15 Nov 2012, 19:03
1
lesnin wrote:
Hi All,

Could I conclude that for this case i.e (1+2x)*x^3*(1-2x)<0
even if one of the terms <0, that does not necessarily mean that the entire product of the 3 terms <0.
Cause like if the eq was (1+2x)*x^3*(1-2x)= 0 ....I could have safely concluded that
However in this case for the entire product <0.. either 1 terms or 2 terms or even all 3 terms can be - ve.

When you have product of two or more terms, the product will be negative only when odd number of terms are negative i.e. either only one term is negative and rest are positive or only 3 terms are negative and rest are positive or only 5 terms are negative and rest are positive.
(-)(+)(+) = (-)
(-)(-)(+) = (+)
(-)(-)(-) = (-)
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Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! Senior Manager Joined: 13 Aug 2012 Posts: 441 Concentration: Marketing, Finance GPA: 3.23 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 06 Dec 2012, 04:15 Using the amazing technique: $$x^3(1-4x^2)<0$$ $$x^3(1-2x)(1+2x)<0$$ + (-1/2) - (0) + (1/2) - If less than 0, select (-) curves. Answer: -1/2 < x < 0 or 1/2 < x ==> C _________________ Impossible is nothing to God. Math Expert Joined: 02 Sep 2009 Posts: 47983 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 17 Jun 2013, 05:52 Bumping for review and further discussion*. Get a kudos point for an alternative solution! *New project from GMAT Club!!! Check HERE _________________ Senior Manager Joined: 13 May 2013 Posts: 430 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 17 Jun 2013, 11:57 This might be a difficult question to answer, but here it is: I understand the methodology in how the correct answer was arrived at (thanks, Bunuel) but my question is, how do I know to use that methodology with this particular question? Also, could I solve for this problem using x^3(1-4x^2)<0 as opposed to (1+2x)*x^3*(1-2x)<0? As always, thanks to the community for all of your help. Manager Joined: 04 Mar 2013 Posts: 76 Location: India Concentration: General Management, Marketing GPA: 3.49 WE: Web Development (Computer Software) Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 02 Jul 2013, 11:17 gmatpapa wrote: Which of the following represents the complete range of x over which x^3 – 4x^5 < 0? A. 0 < |x| < ½ B. |x| > ½ C. –½ < x < 0 or ½ < x D. x < –½ or 0 < x < ½ E. x < –½ or x > 0 my take substitute values 1/2 is in many options try plug in and u find only c, is going correctly, as u can know > 1/2 all work but d also contends , when u look at D first part it leave the race so C Senior Manager Joined: 13 May 2013 Posts: 430 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 09 Jul 2013, 20:15 Which of the following represents the complete range of x over which x^3 – 4x^5 < 0? x^3 – 4x^5 < 0 x^3(1-4x^2) < 0 (1-4x^2) < 0 1 < 4x^2 √1 < √4x^2 (when you take the square root of 4x^2 you take the square root of a square so...) 1 < |2x| 1<(2x) 1/2 < x OR 1<-2x -1/2>x I am still a bit confused as to how we get 0. I see how it is done with the "root" method but my way of solving was just a bit different. Any thoughts? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8195 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 09 Jul 2013, 21:08 1 WholeLottaLove wrote: Which of the following represents the complete range of x over which x^3 – 4x^5 < 0? x^3 – 4x^5 < 0 x^3(1-4x^2) < 0 (1-4x^2) < 0 1 < 4x^2 √1 < √4x^2 (when you take the square root of 4x^2 you take the square root of a square so...) 1 < |2x| 1<(2x) 1/2 < x OR 1<-2x -1/2>x I am still a bit confused as to how we get 0. I see how it is done with the "root" method but my way of solving was just a bit different. Any thoughts? The step in red above is your problem. How did you get rid of x^3? Can you divide both sides by x^3 when you have an inequality? You don't know whether x^3 is positive or negative. If you divide both sides by x^3 and x^3 is negative, the sign will flip. So you must retain the x^3 and that will give you 3 transition points (-1/2, 0 , 1/2) Even in equations, it is not a good idea to cancel off x from both sides. You might lose a solution in that case x = 0 e.g. x(x - 1) = 0 (x - 1) = 0 x = 1 (Incomplete) x(x-1) = 0 x = 0 or 1 (Correct) _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Re: Which of the following represents the complete range of x &nbs [#permalink] 09 Jul 2013, 21:08

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