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# Which of the following represents the complete range of x

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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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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 $$x > 0$$
[Reveal] Spoiler: OA

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

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08 Feb 2011, 09:41
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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
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Re: Which of the following represents the complete range of x [#permalink]

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13 Feb 2011, 22:05
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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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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17827 [23], given: 235 Math Expert Joined: 02 Sep 2009 Posts: 42275 Kudos [?]: 132867 [9], given: 12389 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 10 Feb 2011, 03:28 9 This post received KUDOS Expert's post 25 This post was BOOKMARKED 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. _________________ Kudos [?]: 132867 [9], given: 12389 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17827 [5], given: 235 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 20 Jun 2012, 22:42 5 This post received KUDOS Expert's post 4 This post was BOOKMARKED 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 My Blog Get started with Veritas Prep GMAT On Demand for$199

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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
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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. Kudos [?]: 662 [5], given: 70 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17827 [2], given: 235 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 02 Mar 2011, 19:12 2 This post received KUDOS Expert's post 1 This post was BOOKMARKED 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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02 Aug 2013, 00:21
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VeritasPrepKarishma wrote:
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.
(-)(+)(+) = (-)
(-)(-)(+) = (+)
(-)(-)(-) = (-)

According to me answer should be

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

I solved all the the difficult inequality questions using this technqiue.I guess i am missing something.

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

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23 Mar 2014, 20:52
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Expert's post
seabhi 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

Hi Bunuel,
I tried the trick, however using the equation I am getting different ranges.
below is what I did ..
1) f(x) <0
2) roots are -1/2 , 0, 1/2

- (-1/2) + 0 - 1/2 +
starting from + from right.

now as per this
x< -1/2 and 0<x<1/2

can you advice where I went wrong...

The factors must be of the form (x - a), (x - b) etc. Notice that one factor here is of the form (1 - 2x). You need to change this.

$$(1+2x)*x^3*(1-2x)<0$$
$$2(x + 1/2)*x^3*2(x - 1/2) > 0$$ (note the sign flip)

Now the factors are of the form required and it is clear that the transition points are -1/2, 0, 1/2.

The required range is x > 1/2 or -1/2 < x< 0
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17827 [2], given: 235 Director Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing. Affiliations: University of Chicago Booth School of Business Joined: 03 Feb 2011 Posts: 871 Kudos [?]: 401 [1], given: 123 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 02 Mar 2011, 18:40 1 This post received KUDOS 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 >. Kudos [?]: 401 [1], given: 123 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17827 [1], given: 235 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 15 Nov 2012, 19:03 1 This post received KUDOS Expert's post 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. (-)(+)(+) = (-) (-)(-)(+) = (+) (-)(-)(-) = (-) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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09 Jul 2013, 21:08
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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)
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17827 [1], given: 235 Math Expert Joined: 02 Sep 2009 Posts: 42275 Kudos [?]: 132867 [1], given: 12389 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 18 Jun 2014, 08:44 1 This post received KUDOS Expert's post gauravsoni 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. $$x^3-4x^5<0$$ --> $$x^3(1-4x^2)<0$$ --> $$(1+2x)*x^3*(1-2x)<0$$ --> $$-\frac{1}{2}<x<0$$ or $$x>\frac{1}{2}$$. Answer: C. Check this for more: inequalities-trick-91482.html Hi Bunuel, sorry for this noob question but, can you explain how do you find the sign for the equality roots - (I know how to find the roots but not able to understand how do we equate to the roots) $$-\frac{1}{2}<x<0$$ or $$x>\frac{1}{2}$$. Please read the whole thread and follow the links given in experts posts. You can benefit a lot from this approach. As for your question please read: which-of-the-following-represents-the-complete-range-of-x-108884.html#p868863 Hope this helps. _________________ Kudos [?]: 132867 [1], given: 12389 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17827 [1], given: 235 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 04 Feb 2015, 23:09 1 This post received KUDOS Expert's post VeritasPrepKarishma 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 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 Responding to a pm: Quote: Why we meed to multiply the both sides by -1? What if the question is x^3 ( 2x+1) ( 1-2x )<0 or >0 do we need in this caee to multiply the both sides by -1? We need to bring the factors in the (x - a)(x - b) format instead of (a - x) format. So how do you convert (1 - 2x) into (2x - 1)? You multiply by -1. Say, if you have 1-2x < 0, and you multiply both sides by -1, you get -1*(1 - 2x) > (-1)*0 (note here that the inequality sign flips because you are multiplying by a negative number) -1*(1 - 2x) > (-1)*0 -1 + 2x > 0 (2x -1) > 0 So you converted the factor to x - a form. In case you have x^3 ( 2x+1) ( 1-2x )<0, you will multiply both sides by -1 to get x^3 ( 2x+1) ( 2x - 1 ) > 0 (inequality sign flips) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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22 Feb 2016, 21:30
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Expert's post
nishantdoshi wrote:
VeritasPrepKarishma 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

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

i solved it using similar approach but what i learned from this link : inequalities-trick-91482.html
is that we should plot the number line starting with +ve on the right most segment and then changing the alternatively as we go from right to left but......

using this approach i'm getting

-1/2<x<0 , x>1/2

but there's no option like that its just the opp.

You are right about starting with a positive sign in the rightmost segment. But all factors should be of the form
(ax+b) or (ax - b).
Note that you have 1 - 2x which should be converted to 2x - 1. For that you multiply both sides by -1 and hence the inequality sign flips.
You will get the correct answer.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17827 [1], given: 235 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17827 [1], given: 235 Location: Pune, India Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 11 Jul 2016, 23:10 1 This post received KUDOS Expert's post Anjalika123 wrote: In this case we have points -1/2 ,0, 1/2 So the sequence of signs should be -+-+ So the range should be x<-1/2 or 0<x<1/2 But the OA is different. where did i go wrong ? Recall that if you are going to start with a positive sign from the rightmost region, the factors should be in the form (x - a) etc (a - x) changes the entire thing. x^3(1−2x)(1+2x)<0 has ( 1- 2x) which is 2*(1/2 - x). This is of the form (a - x). You need to multiply the inequality by -1 here to get x^3 * (2x - 1) * (1 + 2x) > 0 Now you will get the correct answer. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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

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

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02 Mar 2011, 18:04
Expert's post
2
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BOOKMARKED
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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Kudos [?]: 132867 [0], given: 12389

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

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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
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Kudos [?]: 132867 [0], given: 12389

Re: Which of the following represents the complete range of x   [#permalink] 04 Mar 2011, 02:52

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