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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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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
[Reveal] Spoiler: OA

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Re: Everything is Less Than Zero [#permalink]

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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 \(x>\frac{1}{2}\).

Answer: C.

Check this for more: inequalities-trick-91482.html
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Re: Everything is Less Than Zero [#permalink]

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

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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: Everything is Less Than Zero [#permalink]

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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 \(x>\frac{1}{2}\).

Answer: C.




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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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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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]

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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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Re: Everything is Less Than Zero [#permalink]

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

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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. :(
please help!!


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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Re: Everything is Less Than Zero [#permalink]

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New post 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: Everything is Less Than Zero [#permalink]

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New post 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: Everything is Less Than Zero [#permalink]

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New post 02 Mar 2011, 18:04
Expert's post
1
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.
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Re: Everything is Less Than Zero [#permalink]

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New post 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 >.
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Re: Everything is Less Than Zero [#permalink]

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New post 04 Mar 2011, 02:52
Expert's post
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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New post 21 Jun 2012, 19:11
thankyou, actually after your reply to the other thread this one i got very easily.
Also the lnk is best for summary as well that you posted.
Re: Which of the following represents the complete range of x   [#permalink] 21 Jun 2012, 19:11

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