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Which of the following represents the complete range of x 08 May 2014, 09:59
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VeritasPrepKarishma

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
Show more

Dear Karishma

So they always have to be in (x-a)(x-B) form???
also, the red part above can be written as \((x+1/2)*x^3*(x-1/2)>0??\)... we can divide both sides by 4 right?? that wont affect the problem na?
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Which of the following represents the complete range of x 08 May 2014, 20:46
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VeritasPrepKarishma

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

Dear Karishma

So they always have to be in (x-a)(x-B) form???
also, the red part above can be written as \((x+1/2)*x^3*(x-1/2)>0??\)... we can divide both sides by 4 right?? that wont affect the problem na?
Show more

Yes, always put them in (x - a)(x - b) format. That way, there will be no confusion.

And yes, you can divide both sides by 4. Think why it doesn't affect our inequality - We have: Expression > 0. If it were 4*Expression, that would be positive too since Expression is positive. If it were Expression/4, that would be positive too since Expression is positive. So positive constants don't affect the inequality.
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Which of the following represents the complete range of x 04 Feb 2015, 23:09
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VeritasPrepKarishma
gmatpapa
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
Show more

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?
Show more

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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Which of the following represents the complete range of x 23 Oct 2015, 13:55
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hi Karshima and Bunuel,
One part of inequality is:(1+2x)<0

How did you guys convert into the answer part: x >-1/2, I am getting x <-1/2...would really appreciate if you could reply. Thanks in advance
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Which of the following represents the complete range of x Updated on: 23 Aug 2023, 00:43
Originally posted by KarishmaB on 29 Oct 2015, 22:21.
Last edited by KarishmaB on 23 Aug 2023, 00:43, edited 1 time in total.
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dep91
hi Karshima and Bunuel,
One part of inequality is:(1+2x)<0

How did you guys convert into the answer part: x >-1/2, I am getting x <-1/2...would really appreciate if you could reply. Thanks in advance
Show more


We cannot take each factor independently. We need to consider all the factors x^3, (1 - 2x) and (1 + 2x) together.
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Which of the following represents the complete range of x 22 Feb 2016, 08:55
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VeritasPrepKarishma
gmatpapa
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
Show more

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!!
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Which of the following represents the complete range of x 22 Feb 2016, 21:30
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nishantdoshi
VeritasPrepKarishma
gmatpapa
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!!
Show more

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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Which of the following represents the complete range of x 09 Jul 2016, 23:37
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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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Anjalika123
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 ?
Show more


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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Which of the following represents the complete range of x 21 Jul 2019, 03:06
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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!
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Which of the following represents the complete range of x 21 Jul 2019, 22:32
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aalakshaya
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!
Show more

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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Which of the following represents the complete range of x 31 Dec 2021, 02:43
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Given that \(x^3 – 4x^5 < 0\) and we need to find the range of values of x

\(x^3 – 4x^5 < 0\)
=> \(x^3 – 4x^3*x^2 < 0\)
=> \(x^3 (1 – 4x^2) < 0\)
=> \(x^3 (1 – (2x)^2) < 0\)
=> \(x*x^2 (1 – 2x) (1+2x) < 0\) [ Using \((a-b)^2 = a^2 - 2ab + b^2\) ]
Multiply both sides by -1 we get
=> \(x*x^2 (2x -1) (2x+1) > 0\)
Now we know that \(x^2\) is always >= 0 so we can ignore \(x^2\) rom above equation
=> \(x* (2x -1) (2x+1) > 0\)

Using Sine Wave method (Watch this video to learn about Sine Wave Method)

We will get the three points as \(\frac{-1}{2}\), 0, \(\frac{1}{2}\)

Attachment:
-0.5 to +0.5 Sine Wave.JPG
-0.5 to +0.5 Sine Wave.JPG [ 18.28 KiB | Viewed 2119 times ]

Since question is asking for > 0 so we will take the "+" range in the above figure

=> \(\frac{-1}{2}\) < x < 0 and x > \(\frac{1}{2}\)

So, Answer will be C
Hope it helps!

Watch the following video to learn How to Solve Inequality Problems

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Which of the following represents the complete range of x 30 Apr 2022, 11:38
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KarishmaB
I have been tying hard to find a detailed explanation of this concept on google to understadn the mechanics intuitively however I have not been been able to. Could you pls share the relavant link explaining this in detail. COuld be a blog or video. Also will appreciate if you can help with other queries too.

KarishmaB
subhashghosh
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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Which of the following represents the complete range of x Updated on: 23 Aug 2023, 00:44
Originally posted by KarishmaB on 01 May 2022, 23:41.
Last edited by KarishmaB on 23 Aug 2023, 00:44, edited 1 time in total.
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Here is a post discussing application of inequalities and absolute values: https://anaprep.com/algebra-inequalitie ... in-action/


ag153
KarishmaB
I have been tying hard to find a detailed explanation of this concept on google to understadn the mechanics intuitively however I have not been been able to. Could you pls share the relavant link explaining this in detail. COuld be a blog or video. Also will appreciate if you can help with other queries too.

KarishmaB
subhashghosh
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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Which of the following represents the complete range of x 02 May 2022, 00:02
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Asked: Which of the following represents the complete range of x over which \(x^3 – 4x^5 < 0\)?

\(xˆ3 - 4xˆ5 < 0\)
\(xˆ3 (1 - 4xˆ2) < 0\)
\(xˆ3 (1 - 2x) (1+2x) < 0\)
\(xˆ3 (x - 1/2)(x + 1/2) > 0\)

-1/2 < x < 0 & x > 1/2

IMO C
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Which of the following represents the complete range of x 10 Oct 2022, 11:27
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Bunuel
144144
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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Hi,

While I partially understand the wavy line method to solve inequalities, can you help me understand why do we equate the expressions to zero to get the roots? This is an inequality right? And since it is <0, one of them needs to be negative and 2 of them positive. Please help me fill the gap here.

Bunuel chetan2u KarishmaB
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Which of the following represents the complete range of x 10 Oct 2022, 21:51
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prakashb2497
Bunuel
144144
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.



Hi,

While I partially understand the wavy line method to solve inequalities, can you help me understand why do we equate the expressions to zero to get the roots? This is an inequality right? And since it is <0, one of them needs to be negative and 2 of them positive. Please help me fill the gap here.

Bunuel chetan2u KarishmaB
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I have explained it here: https://gmatclub.com/forum/inequalities ... ml#p804990
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suhaanimanektala
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Which of the following represents the complete range of x 10 Jul 2025, 20:55
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KarishmaB, in the second bracket, wouldn't we factor out -2 so that we're left with (x-1/2) and THEN we flip the sign?
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Which of the following represents the complete range of x 13 Jul 2025, 22:09
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It's the same thing. You can factor out -2 and then remove the negative sign and flip the inequality sign.
or you can factor out 2 and multiply both sides by -1 which will flip the inequality sign. Both moves lead to the same thing.

suhaanimanektala
KarishmaB, in the second bracket, wouldn't we factor out -2 so that we're left with (x-1/2) and THEN we flip the sign?
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