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Which of the following represents the complete range of x

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

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New post 18 Jun 2016, 00:44
Found the method to solve inequalities at inequalities-trick-91482.html very useful. Applying it to the questinon above

x^3 - 4x^5 <0
x^3(1 - 4x^2) <0
x3(1−2x)(1+2x)<0
This gives 3 roots - 0 (by equating x3=0), x=1/2 (by equating 1-2x=0) and x=-1/2 (by equating 1+2x=0)

On a number line, we have 4 regions

-------- -1/2 -------- 0 --------- 1/2 -------

I used -1, -1/3, 1/3 and 1 as data sets for each region and put them in x3(1−2x)(1+2x) eq.
For x=-1, x3(1−2x)(1+2x) is a +ve expression (-1*3*-1=3). So function is +vw for x< -1/2 --- range1
For x=-1/3, x3(1−2x)(1+2x) is a -ve expression (-1/27*5/3*1/3). So fn is -ve for 1/2<=x<0 ------ range 2
For x=1/3, x3(1−2x)(1+2x) is +ve. So fn is +ve for 0<=x<1/2------ range 3
For x=1, x3(1−2x)(1+2x) is -ve. So fn is -ve for 1/2<=x------ range 4

The original expression (condition) is x^3 - 4x^5 <0. So we are interested in -ve function only which are given by ranges 2 and 4 only. Thus answer is 1/2<=x<0 and 1/2<=x which is same as choice C (–½ < x < 0 or ½ < x)
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 09 Jul 2016, 23:37
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 ?
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 11 Jul 2016, 23:10
1
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.
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 25 Jul 2016, 20:13
x^3 – 4x^5 < 0
x^3 < 4x^5
x * x * x < 4 * x * x * x * x * x
Cancelling both sides... 1 < 4 * x * x
1/4 < x^2
sqrt(1/4) < x...
so, –½ < x or ½ < x.
there is a tricky situation here where x should be more than –½, but x should also be more than ½.
But we know 0 cannot be an option, as 0 - 0 is not < 0.
Hence, option C..
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 08 Oct 2016, 06:13
Bunuel wrote:
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.



when should we include zero in the range ....please help me understand because here : inequalities-trick-91482.html @fluke's solution doesnt contain 0 in the set of ranges. :/

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

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New post 08 Oct 2016, 06:50
nishantdoshi wrote:
Bunuel wrote:
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.



when should we include zero in the range ....please help me understand because here : inequalities-trick-91482.html @fluke's solution doesnt contain 0 in the set of ranges. :/

HELP


We should include 0 in the range when the equation is of the form x^3 - 4x^5 =< 0.

Notice the sign of the inequality. We have less than and equal to.
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 08 Oct 2016, 06:59
and here in this partivular question we take 0 in the range because we get 0 as one of the roots...right?
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New post 08 Oct 2016, 07:06
nishantdoshi wrote:
and here in this partivular question we take 0 in the range because we get 0 as one of the roots...right?


−1/2 <x<0 doesn't mean 0 is in the range. It means x could be anything less than 0 but greater than -1/2.

Had the solution been −1/2 =<x=<0, we would have said 0 is in the range.
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 07 Aug 2017, 15:23
we transform the inequality to x^3 ( 1-2x)(1+2x)<0
roots and key points are -½, 0 and ½
So 4 zones on the line starting from negative as there is a negative x in one of the inequality terms
+ - + -
--------------(-½)--------0-----------(½)------------->
So our range is:
-½<x<0 and x>½
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 19 Mar 2018, 12:14
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 >.


Wow!!! My entire life is a myth<.> :shocked :shocked
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Which of the following represents the complete range of x  [#permalink]

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New post 08 Jan 2019, 21:21
================================
after expanding the equation it becomes
(1+2x)*x^3*(1-2x)<0
x^3<0 gives x<0...first case
1+2x<0 gives x<-1/2..second case
1-2x<0 gives x>1/2...third case


not sure why you have taken x>-1/2, which is opposite to the 2nd case.
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 19 Jun 2019, 03:32
It's important to remember that the positive rightmost mark is applicable and exact ONLY when the value of x has to be positive.

However, in this case
x^3 - 4x^5 < 0
--->x^3(1-2x)(1+2x) < 0
---> -x^3(2x-1)(1+2x) < 0 (I put negative sign in order to switch the value of x from (1-2x) into positive)
--> x^3(2x-1)(1+2x) > 0 (multiply by -1)
From there, we can draw the number line and choose the regions that give positive values
--->So, -(1/2) < x < 0, x> (1/2)--->C
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 21 Jul 2019, 03:06
Hi Bunuel VeritasKarishma , I solved this question in this manner:-


1. x^3 - 4x^2<0
= x^3<4x^3

Now, when x>0

dividing both sides by x^3
1<4x^2 which means x>(1/2)

in the same way, when x<0

dividing both sides by x^3
1>4x^2 which means x<(1/2) but since x is negative, the range shall be x<0

from both these ranges, my final answer was x>(1/2) and x<0

Could you please tell me what is wrong in my approach?
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 21 Jul 2019, 22:32
aalakshaya wrote:
Hi Bunuel VeritasKarishma , I solved this question in this manner:-


1. x^3 - 4x^2<0
= x^3<4x^3

Now, when x>0

dividing both sides by x^3
1<4x^2 which means x>(1/2)

in the same way, when x<0

dividing both sides by x^3
1>4x^2 which means x<(1/2) but since x is negative, the range shall be x<0

from both these ranges, my final answer was x>(1/2) and x<0

Could you please tell me what is wrong in my approach?
Thank you!


Given x^2 = a^2, when you take square root, you get |x| = a so x can be a or -a.

\(x^2 > \frac{1}{4}\)

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

So \(x > \frac{1}{2}\) or \(x < \frac{-1}{2}\)

Since x >= 0, it must be x > 1/2



\(x^2 < \frac{1}{4}\)

\(|x| < \frac{1}{2}\)

\(\frac{-1}{2} < x < \frac{1}{2}\)

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

So x > 1/2 OR -1/2 < x< 0
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Re: Which of the following represents the complete range of x  [#permalink]

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New post 12 Nov 2019, 09:42
Bunuel wrote:
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.


Bunuel

Thanks for the link, the trick-thread is definitely helpful.

Now using it for this problem: Considering finding the + and negative range after I know -1/2, 0, +1/2. Do I always need to look for extreme values to see whether the cycle "starts" with plus or minus? Or is there a general rule such that if the number of roots is odd (like in this case) it starts neg. and ends pos. and whenever the number of roots is even, it starts as well as ends positive?

Thanks in advance, I hope I formulated the question in an more or less understandable way :-D
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Re: Which of the following represents the complete range of x   [#permalink] 12 Nov 2019, 09:42

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