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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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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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10 Feb 2011, 03:28
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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.
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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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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 [?]: 17346 [22], given: 232 Math Expert Joined: 02 Sep 2009 Posts: 41885 Kudos [?]: 128707 [0], given: 12182 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 02 Mar 2011, 18:04 Expert's post 2 This post was 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 $$x>\frac{1}{2}$$. Answer: C. 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. _________________ Kudos [?]: 128707 [0], given: 12182 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 [?]: 396 [0], given: 123 Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 02 Mar 2011, 18:40 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 [?]: 396 [0], given: 123 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7674 Kudos [?]: 17346 [2], given: 232 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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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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Re: Which of the following represents the complete range of x [#permalink]

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20 Jun 2012, 22:42
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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

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
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17346 [5], given: 232 VP Status: Final Lap Up!!! Affiliations: NYK Line Joined: 21 Sep 2012 Posts: 1077 Kudos [?]: 646 [5], given: 70 Location: India GMAT 1: 410 Q35 V11 GMAT 2: 530 Q44 V20 GMAT 3: 630 Q45 V31 GPA: 3.84 WE: Engineering (Transportation) Re: Which of the following represents the complete range of x [#permalink] ### Show Tags 01 Nov 2012, 16:06 5 This post received KUDOS Bunuel Thanx a trillion for your post on solving inequalities using graph You know i paid over 300$ 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......
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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
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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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Re: Which of the following represents the complete range of x   [#permalink] 09 Jul 2013, 21:08

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