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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 17 Jun 2016, 23:44
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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, 22:37
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, 22:10
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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, 19:13
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, 05:13
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, 05:50
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, 05:59
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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Re: Which of the following represents the complete range of x [#permalink]

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New post 08 Oct 2016, 06:06
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, 14:23
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
Re: Which of the following represents the complete range of x   [#permalink] 07 Aug 2017, 14:23

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