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Math Expert V
Joined: 02 Sep 2009
Posts: 59634
Re: Which of the following represents the complete range of x  [#permalink]

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

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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 V
Joined: 16 Oct 2010
Posts: 9866
Location: Pune, India
Re: Which of the following represents the complete range of x  [#permalink]

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

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

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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 GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9866
Location: Pune, India
Re: Which of the following represents the complete range of x  [#permalink]

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2
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
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Karishma
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Re: Which of the following represents the complete range of x  [#permalink]

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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?
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
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Re: Which of the following represents the complete range of x  [#permalink]

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

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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 V
Joined: 02 Sep 2009
Posts: 59634
Re: Which of the following represents the complete range of x  [#permalink]

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1
1
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.
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Re: Which of the following represents the complete range of x  [#permalink]

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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 V
Joined: 16 Oct 2010
Posts: 9866
Location: Pune, India
Re: Which of the following represents the complete range of x  [#permalink]

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1
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)
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Re: Which of the following represents the complete range of x  [#permalink]

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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 V
Joined: 02 Sep 2009
Posts: 59634
Re: Which of the following represents the complete range of x  [#permalink]

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

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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 1390 times ]

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

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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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Joined: 16 Oct 2010
Posts: 9866
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Re: Which of the following represents the complete range of x  [#permalink]

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dep91 wrote:
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

We cannot take each factor independently. We need to consider all the factors x^3, (1 - 2x) and (1 + 2x) together.

Here are 3 posts that will help you understand this concept in detail:
http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
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Karishma
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Re: Which of the following represents the complete range of x  [#permalink]

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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

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. Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9866
Location: Pune, India
Re: Which of the following represents the complete range of x  [#permalink]

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1
nishantdoshi wrote:
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

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. 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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Karishma
Veritas Prep GMAT Instructor

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Posts: 6
Re: Which of the following represents the complete range of x  [#permalink]

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

--------------------------------------------------------

Hi Bunuel,

Why we cant do like this

x3<4x5

divide by x3

1<4x2

for this equation

-1/2<x<0 doesn't hold true Re: Which of the following represents the complete range of x   [#permalink] 14 Jun 2016, 20:29

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