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Manager  Joined: 10 May 2009
Posts: 52
How many roots does the equation || x +2 | - 2 | = 2 have?  [#permalink]

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39 00:00

Difficulty:   65% (hard)

Question Stats: 53% (01:37) correct 47% (01:36) wrong based on 807 sessions

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How many roots does the equation || x +2 | - 2 | = 2 have?

A. 0
B. 1
C. 2
D. 3
E. 4

M13-02

Originally posted by prinits on 07 Jun 2009, 02:13.
Last edited by Bunuel on 15 Mar 2014, 05:38, edited 2 times in total.
Renamed the topic and edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58098
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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6
8
imhimanshu wrote:
Hi Experts,
I would like to understand what exactly the question is testing? What are we supposed to find in such questions.

Thanks
H

We should find the number of values of x that satisfy the given equation.

How many roots does the equation || x +2 | - 2 | = 2 have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$||x +2|-2|=2$$ two cases:

1. $$|x +2|=0$$ --> $$x=-2$$;
2. $$|x +2|=4$$ --> $$x=2$$ or $$x=-6$$.

So, the given equation has three roots: -2, 2, and -6.

_________________
VP  Joined: 04 May 2006
Posts: 1454
Schools: CBS, Kellogg
Re: Roots issue  [#permalink]

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12
2
prinits wrote:
How many roots does this equation have?

||x+2|-2|=2

the same as:|x+2|-2 = + - 2

1. |x+2|-2=2 -->|x+2|=4, again, x+2=+ -4
1.1, x+2=4, x=2
1.2, x+2=-4, x=-6

2. |x+2|-2=-2 -->|x+2|=0, x=-2

So, x=-6,-2,2
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##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 58098
Re: M13-2 GMATCLUB MATH Q  [#permalink]

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4
1
WoundedTiger wrote:
Q. How many roots does the equation ||x+2|−2|=2 have?

a. 0
b. 1
c. 2
d. 3
3. 4

I am okay with the solution given for the above problem but I tried solving by solving algebraically and I missed out on one of the roots.

||x+2|−2|=2----->>Can be re-written as

$$\sqrt{({| {|x+2|}−{2}|)^{2}}$$ = 2
Squaring both sides we get
(||x+2|−2|)^2 = 4
Now |x+2| is positive so the expression can be written as ((x+2)-2)^2= 4 or x^2=4 or x=-2 or 2

How do I get -6 (another root) which should be there actually...Did I do anything wrong above.

PS: Looked on the forum before but could not find anything so posting it here.

Yes |x+2| is positive but |x+2| = x+2, only when x>=-2. What you should have done is as follows:

Square: $$||x+2|-2|=2$$:

$$(x+2)^2-4|x+2|+4=4$$;

$$x^2+4x+4-4|x+2|+4=4$$;

$$x^2+4x+4=4|x+2|$$;

$$(x+2)^2=4|x+2|$$;

$$(x+2)^4=16(x+2)^2$$;

$$((x+2)^2-4(x+2))((x+2)^2-4(x+2))=0$$;

$$(x^2-4)(x+2)(x+6)=0$$;

$$x=2$$, $$x=-2$$, or $$x=-6$$.

Hope it's clear.
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58098
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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3
1
imhimanshu wrote:
Bunuel wrote:

We should find the number of values of x that satisfy the given equation.

Thanks Bunuel for the explanation. However, I would like to ask, Is it possible to solve it through graphs. Also, What does above equation mean in terms of graphical representation. i.e Do we need to find the number of points for line y=2 intersects the graph. Is it correct representation of the given equation.

Regards,
H

Possible? Yes.
Good approach? No.

As for the graph itself:
Attachment: graph.png [ 5.93 KiB | Viewed 12798 times ]
Hope it helps.
_________________
Manager  Joined: 03 Jan 2015
Posts: 59
Concentration: Strategy, Marketing
WE: Research (Pharmaceuticals and Biotech)
Re: How many roots does the equation || x +2 | - 2 | = 2 have?  [#permalink]

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3
||x + 2 |-2| = 2
|x + 2| - 2 = 2 or |x + 2| - 2 = -2
|x+2| = 4 or |x + 2| = 0
x + 2 = 4 or x + 2 = -4 or x + 2 = 0
x = 2 or x = -6 or x = -2

Math Expert V
Joined: 02 Sep 2009
Posts: 58098
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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1
PathFinder007 wrote:
Bunuel wrote:
imhimanshu wrote:
Hi Experts,
I would like to understand what exactly the question is testing? What are we supposed to find in such questions.

Thanks
H

We should find the number of values of x that satisfy the given equation.

How many roots does the equation || x +2 | - 2 | = 2 have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$||x +2|-2|=2$$ two cases:

1. $$|x +2|=0$$ --> $$x=-2$$;
2. $$|x +2|=4$$ --> $$x=2$$ or $$x=-6$$.

So, the given equation has three roots: -2, 2, and -6.

Hi Bunnel,

I have one doube here how are you getting second equation because ? are we transfering -2 from lhs to RHS. but this is an absolute value so how can we transfer this. it will become 0 in RHS.

Thanks

No, I'm not doing that.

Let |x +2|=a, then we'd have: $$|a-2|=2$$:

1. $$a=0$$ --> $$|x +2|=0$$ --> $$x=-2$$;
2. $$a=4$$ --> $$|x +2|=4$$ --> $$x=2$$ or $$x=-6$$.

Hope it's clear.
_________________
EMPOWERgmat Instructor V
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: How many roots does the equation || x +2 | - 2 | = 2 have?  [#permalink]

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1
Hi All,

Since this equation has an absolute value embedded inside another absolute value, the key to solving it is to work from the "outside" in…

We're given || X +2 | - 2 | = 2

Let's rewrite this as…

||something| - 2| = 2

Taking the "-2" and the "outside" absolute value into account, we need the left "side" of the equation to equal EITHER 2 or -2….

So….

|something| - 2 = 2
|something| - 2 = -2

In the first option, we need |something| to equal 4
In the second option, we need |something| to equal 0

Now we can deal with the inner absolute value….

|X + 2| = 4

This has two solutions: -6 and +2

|X + 2| = 0

This has one solution: -2

Thus, we have 3 solutions/roots.

GMAT assassins aren't born, they're made,
Rich
_________________
Senior Manager  Joined: 07 Sep 2010
Posts: 251
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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Hi Experts,
I would like to understand what exactly the question is testing? What are we supposed to find in such questions.

Thanks
H
Senior Manager  Joined: 07 Sep 2010
Posts: 251
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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Bunuel wrote:

We should find the number of values of x that satisfy the given equation.

Thanks Bunuel for the explanation. However, I would like to ask, Is it possible to solve it through graphs. Also, What does above equation mean in terms of graphical representation. i.e Do we need to find the number of points for line y=2 intersects the graph. Is it correct representation of the given equation.

Regards,
H
Senior Manager  Joined: 13 May 2013
Posts: 410
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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Could you explain this one to me? Why can we just drop the outer absolute value signs? Why don't we do the same for the inner ones? What happens if this is an inequality with three pairs of absolute value signs in one another?

Thanks!

Bunuel wrote:
imhimanshu wrote:
Hi Experts,
I would like to understand what exactly the question is testing? What are we supposed to find in such questions.

Thanks
H

We should find the number of values of x that satisfy the given equation.

How many roots does the equation || x +2 | - 2 | = 2 have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$||x +2|-2|=2$$ two cases:

1. $$|x +2|=0$$ --> $$x=-2$$;
2. $$|x +2|=4$$ --> $$x=2$$ or $$x=-6$$.

So, the given equation has three roots: -2, 2, and -6.

Director  Joined: 25 Apr 2012
Posts: 665
Location: India
GPA: 3.21
WE: Business Development (Other)
M13-2 GMATCLUB MATH Q  [#permalink]

### Show Tags

Q. How many roots does the equation ||x+2|−2|=2 have?

a. 0
b. 1
c. 2
d. 3
3. 4

I am okay with the solution given for the above problem but I tried solving by solving algebraically and I missed out on one of the roots.

||x+2|−2|=2----->>Can be re-written as

$$\sqrt{({| {|x+2|}−{2}|)^{2}}$$ = 2
Squaring both sides we get
(||x+2|−2|)^2 = 4
Now |x+2| is positive so the expression can be written as ((x+2)-2)^2= 4 or x^2=4 or x=-2 or 2

How do I get -6 (another root) which should be there actually...Did I do anything wrong above.

PS: Looked on the forum before but could not find anything so posting it here.

_________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”
Director  Joined: 25 Apr 2012
Posts: 665
Location: India
GPA: 3.21
WE: Business Development (Other)
Re: M13-2 GMATCLUB MATH Q  [#permalink]

### Show Tags

Bunuel wrote:
WoundedTiger wrote:
Q. How many roots does the equation ||x+2|−2|=2 have?

a. 0
b. 1
c. 2
d. 3
3. 4

I am okay with the solution given for the above problem but I tried solving by solving algebraically and I missed out on one of the roots.

||x+2|−2|=2----->>Can be re-written as

$$\sqrt{({| {|x+2|}−{2}|)^{2}}$$ = 2
Squaring both sides we get
(||x+2|−2|)^2 = 4
Now |x+2| is positive so the expression can be written as ((x+2)-2)^2= 4 or x^2=4 or x=-2 or 2

How do I get -6 (another root) which should be there actually...Did I do anything wrong above.

PS: Looked on the forum before but could not find anything so posting it here.

Yes |x+2| is positive but |x+2| = x+2, only when x>=-2. What you should have done is as follows:

Square: $$||x+2|-2|=2$$:

$$(x+2)^2-4|x+2|+4=4$$;

$$x^2+4x+4-4|x+2|+4=4$$;

$$x^2+4x+4=4|x+2|$$;

$$(x+2)^2=4|x+2|$$;

$$(x+2)^4=16(x+2)^2$$;

$$((x+2)^2-4(x+2))((x+2)^2-4(x+2))=0$$;

$$(x^2-4)(x+2)(x+6)=0$$;

$$x=2$$, $$x=-2$$, or $$x=-6$$.

Hope it's clear.

Thanks for the solution Bunuel!!!

This one is certainly complicated if we follow algebra.
_________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”
Manager  B
Joined: 10 Mar 2014
Posts: 182
Re: How many roots does this equation have? ||x+2|-2|=2  [#permalink]

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Bunuel wrote:
imhimanshu wrote:
Hi Experts,
I would like to understand what exactly the question is testing? What are we supposed to find in such questions.

Thanks
H

We should find the number of values of x that satisfy the given equation.

How many roots does the equation || x +2 | - 2 | = 2 have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$||x +2|-2|=2$$ two cases:

1. $$|x +2|=0$$ --> $$x=-2$$;
2. $$|x +2|=4$$ --> $$x=2$$ or $$x=-6$$.

So, the given equation has three roots: -2, 2, and -6.

Hi Bunnel,

I have one doube here how are you getting second equation because ? are we transfering -2 from lhs to RHS. but this is an absolute value so how can we transfer this. it will become 0 in RHS.

Thanks
Intern  Joined: 29 Sep 2014
Posts: 2
Concentration: Marketing, General Management
GMAT Date: 04-30-2015
Re: How many roots does the equation || x +2 | - 2 | = 2 have?  [#permalink]

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1
We should find the number of values of x that satisfy the given equation.

How many roots does the equation || x +2 | - 2 | = 2 have?

A. 0
B. 1
C. 2
D. 3
E. 4

||x+2|−2|=2 two cases:

1. |x+2|=0 --> x=−2;
2. |x+2|=4 --> x=2 or x=−6.

So, the given equation has three roots: -2, 2, and -6.

Still not clear how to determine " two cases/ three cases" in general for all such equations .
The GMAT book also talks about key points and associated condition in the three stage approach for inequalities calculations
Non-Human User Joined: 09 Sep 2013
Posts: 12523
Re: How many roots does the equation || x +2 | - 2 | = 2 have?  [#permalink]

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_________________ Re: How many roots does the equation || x +2 | - 2 | = 2 have?   [#permalink] 17 Apr 2019, 00:07
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