Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 22 Oct 2016, 01:56

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Which of the following represents the complete range of x

Author Message
TAGS:

Hide Tags

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6962
Location: Pune, India
Followers: 2024

Kudos [?]: 12717 [1] , given: 221

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

Show Tags

09 Jul 2013, 21:08
1
KUDOS
Expert's post
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)
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 13 May 2013 Posts: 472 Followers: 3 Kudos [?]: 146 [0], given: 134 Re: Which of the following represents the complete range of x [#permalink] Show Tags 09 Jul 2013, 22:10 Thank you, I figured that was my problem and you confirmed it! VeritasPrepKarishma wrote: 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) Math Expert Joined: 02 Sep 2009 Posts: 35244 Followers: 6619 Kudos [?]: 85353 [0], given: 10236 Re: Which of the following represents the complete range of x [#permalink] Show Tags 09 Jul 2013, 22:40 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? You cannot reduce by x^3 in the red part. _________________ Senior Manager Joined: 13 May 2013 Posts: 472 Followers: 3 Kudos [?]: 146 [0], given: 134 Re: Which of the following represents the complete range of x [#permalink] Show Tags 09 Jul 2013, 22:43 I didn't. I was trying to show that x^3 < 0 and (1-4x^2) < 0 (to get the check points) but I forgot to show for x^3! Thank you, though! Bunuel wrote: 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? You cannot reduce by x^3 in the red part. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6962 Location: Pune, India Followers: 2024 Kudos [?]: 12717 [0], given: 221 Re: Which of the following represents the complete range of x [#permalink] Show Tags 01 Aug 2013, 23:29 WholeLottaLove wrote: 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. 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. For more, check: http://www.veritasprep.com/blog/2012/07 ... s-part-ii/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Manager
Joined: 01 Jan 2013
Posts: 67
Location: India
Followers: 0

Kudos [?]: 26 [2] , given: 131

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

Show Tags

02 Aug 2013, 00:21
2
KUDOS
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.
Manager
Joined: 22 Aug 2013
Posts: 113
Schools: ISB '15
Followers: 2

Kudos [?]: 26 [0], given: 60

Re: Everything is Less Than Zero [#permalink]

Show Tags

21 Mar 2014, 05:59
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 $$x>\frac{1}{2}$$.

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...
_________________

Veritas Prep - 650
MGMAT 1 590
MGMAT 2 640 (V48/Q31)

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6962
Location: Pune, India
Followers: 2024

Kudos [?]: 12717 [2] , given: 221

Re: Everything is Less Than Zero [#permalink]

Show Tags

23 Mar 2014, 20:52
2
KUDOS
Expert's post
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 $$x>\frac{1}{2}$$.

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
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 22 Aug 2013 Posts: 113 Schools: ISB '15 Followers: 2 Kudos [?]: 26 [0], given: 60 Re: Which of the following represents the complete range of x [#permalink] Show Tags 23 Mar 2014, 21:33 Thanks Karishma. +1 kudos.. _________________ Veritas Prep - 650 MGMAT 1 590 MGMAT 2 640 (V48/Q31) Please help the community by giving Kudos. Manager Joined: 20 Oct 2013 Posts: 66 Followers: 0 Kudos [?]: 2 [0], given: 27 Re: Everything is Less Than Zero [#permalink] Show Tags 08 May 2014, 09:59 VeritasPrepKarishma wrote: 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? _________________ Hope to clear it this time!! GMAT 1: 540 Preparing again Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6962 Location: Pune, India Followers: 2024 Kudos [?]: 12717 [0], given: 221 Re: Everything is Less Than Zero [#permalink] Show Tags 08 May 2014, 20:46 nandinigaur wrote: VeritasPrepKarishma wrote: 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? 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Intern
Joined: 05 Feb 2014
Posts: 48
Followers: 0

Kudos [?]: 13 [0], given: 49

Re: Everything is Less Than Zero [#permalink]

Show Tags

18 Jun 2014, 07:09
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 $$x>\frac{1}{2}$$.

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 $$x>\frac{1}{2}$$.
Math Expert
Joined: 02 Sep 2009
Posts: 35244
Followers: 6619

Kudos [?]: 85353 [1] , given: 10236

Re: Everything is Less Than Zero [#permalink]

Show Tags

18 Jun 2014, 08:44
1
KUDOS
Expert's post
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 $$x>\frac{1}{2}$$.

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 $$x>\frac{1}{2}$$.

Hope this helps.
_________________
Current Student
Joined: 20 Jan 2014
Posts: 186
Location: India
Concentration: Technology, Marketing
Followers: 1

Kudos [?]: 47 [0], given: 120

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

Show Tags

25 Sep 2014, 05:34
Just learned this technique from one of the Bunuel's post - He is the Champ.

x^3 - 4x^5 < 0?

X^3(1-4X^2) < 0

X^3 < 0 => x<0
1-4X^2 < 0 => X< +- 1/2

Now write all factors in ascending order:
(X+1/2)(X-0)(X-1/2) < 0

No Using excellent property described here : inequalities-trick-91482.html

+____ -1/2 ____ - _____0 ____ +____ 1/2 _____ - ___

the negative parts are -1/2 < X < 0 and X> 1/2
_________________

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6962
Location: Pune, India
Followers: 2024

Kudos [?]: 12717 [1] , given: 221

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

Show Tags

04 Feb 2015, 23:09
1
KUDOS
Expert's post
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)
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Intern
Joined: 03 Jul 2015
Posts: 40
Followers: 0

Kudos [?]: 10 [0], given: 27

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

Show Tags

03 Sep 2015, 21:29
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 $$x>\frac{1}{2}$$.

Check this for more: inequalities-trick-91482.html

why dont we reduce x^3 from both side??
Math Expert
Joined: 02 Sep 2009
Posts: 35244
Followers: 6619

Kudos [?]: 85353 [0], given: 10236

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

Show Tags

03 Sep 2015, 23:20
anik19890 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 $$x>\frac{1}{2}$$.

Check this for more: inequalities-trick-91482.html

why dont we reduce x^3 from both side??

When reducing an inequality by a variable, we must know it's sign: if it's positive the sign of the inequality stays the same but if it's negative the sign of the inequality flips.
_________________
Director
Joined: 10 Mar 2013
Posts: 608
Location: Germany
Concentration: Finance, Entrepreneurship
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
Followers: 11

Kudos [?]: 211 [0], given: 200

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

Show Tags

19 Oct 2015, 13:21
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

1. $$x^3(1-4x^2)=x^3*(1-2x)(1+2x)<0$$
2. Find critical points (zero points): x=0, x=1/2, x=-1/2
3. Pick some easy numbers to test each range (see attachment) and insert it in the inequality to find the final sign of it:
-1: - + - +
-1/4: - + + -
1/4: + ++ +
1: + - + -

To answer this question we just need describe the range with a -ve sign from the attachment: $$-\frac{1}{2}< X < 0$$ and x > $$\frac{1}{2}$$
Attachments

Inequality.png [ 2.5 KiB | Viewed 386 times ]

_________________

When you’re up, your friends know who you are. When you’re down, you know who your friends are.

800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50
GMAT PREP 670
MGMAT CAT 630
KAPLAN CAT 660

Intern
Joined: 04 Oct 2015
Posts: 2
Followers: 0

Kudos [?]: 0 [0], given: 11

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

Show Tags

23 Oct 2015, 13:55
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
Manager
Joined: 23 Sep 2015
Posts: 95
Concentration: General Management, Finance
GMAT 1: 680 Q46 V38
GMAT 2: 690 Q47 V38
GPA: 3.5
Followers: 0

Kudos [?]: 12 [0], given: 213

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

Show Tags

29 Oct 2015, 21:34
thanks, figured my own question out -edit.
Which of the following represents the complete range of x   [#permalink] 29 Oct 2015, 21:34

Go to page   Previous    1   2   3    Next  [ 52 posts ]

Similar topics Replies Last post
Similar
Topics:
1 Which of the following represents the profit from the investment of x 5 23 Jul 2015, 03:22
4 If x + y= 8z, then which of the following represents the ave 5 19 Feb 2014, 01:21
34 Which of the following represents 1<x<9? 9 31 Jan 2012, 11:22
13 Which of the following inequalities have a finite range of values of x 14 03 Sep 2010, 07:09
2 If x+y=8z, then which of the following represents the averag 5 08 Feb 2011, 03:40
Display posts from previous: Sort by