Which of the following represents the complete range of x

Question

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 $x > 0$

Bunuel · Accepted Answer

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}\frac{1}{2} . Answer: C. Check this for more: https://gmatclub.com/forum/inequalities-trick-91482.html

Bunuel · Answer

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}\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}\frac{1}{2} (4th range). Hope its clear.

KarishmaB · Answer

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 >. Check out this post for more on complete range questions: https://anaprep.com/algebra-must-be-tru ... questions/

Archit143 · Answer

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.

144144 · Answer

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.

subhashghosh · Answer

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

Bunuel · Answer

9. Inequalities 
     Theory   
 Inequalities Made Easy! 
 Inequalities: Tips and hints 
 Arithmetic with Inequalities 
 Wavy Line Method Application - Complex Algebraic Inequalities 
 Solving Quadratic Inequalities: Graphic Approach 
 Inequalities - Quadratic Inequalities 
 Graphic approach to problems with inequalities 
 Inequalities trick 
 INEQUATIONS(Inequalities) Part 1 
 INEQUATIONS(Inequalities) Part 2 
     Questions   
 Inequality and absolute value questions from my collection 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

gmat1220 · Answer

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.

KarishmaB · Answer

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.

KarishmaB · Answer

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 Now plot them on the number line and get the regions where this inequality holds.

lesnin · Answer

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.

KarishmaB · Answer

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.
(-)(+)(+) = (-)
(-)(-)(+) = (+)
(-)(-)(-) = (-)

mbaiseasy · Answer

Using the amazing technique:

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

WholeLottaLove · Answer

This might be a difficult question to answer, but here it is:

I understand the methodology in how the correct answer was arrived at (thanks, Bunuel) but my question is, how do I know to use that methodology with this particular question?

Also, could I solve for this problem using x^3(1-4x^2)<0 as opposed to (1+2x)*x^3*(1-2x)<0?

As always, thanks to the community for all of your help.

WholeLottaLove · Answer

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?

KarishmaB · Answer

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)

KarishmaB · Answer

When you have linear factors and inequalities, think of this method. Since this method is useful for linear factors, you need to split the quadratic (1 - 4x^2) into (1-2x)*(1+2x).
Some quadratic or higher powers may not be a problem (e.g. (x + 1)^2, (x^2 + 1) etc are always positive) so they can be ignored.

seabhi · Answer

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...

KarishmaB · Answer

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

Which of the following represents the complete range of x

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Which of the following represents the complete range of x

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