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

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How many roots does the equation || x +2 | - 2 | = 2 have? [#permalink]

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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.
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Re: Roots issue [#permalink]

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New post 07 Jun 2009, 02:21
11
1
prinits wrote:
How many roots does this equation have?

||x+2|-2|=2

Please explain!


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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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 29 Apr 2013, 07:13
Hi Experts,
I would like to understand what exactly the question is testing? What are we supposed to find in such questions.

Thanks
H
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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 29 Apr 2013, 07:26
3
7
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.

Answer: D.
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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 30 Apr 2013, 05:19
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.
Please clarify.

Regards,
H
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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 30 Apr 2013, 06:26
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.
Please clarify.

Regards,
H


Possible? Yes.
Good approach? No.

As for the graph itself:
Attachment:
graph.png
graph.png [ 5.93 KiB | Viewed 9836 times ]
Hope it helps.
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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 24 Jun 2013, 13:06
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.

Answer: D.
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M13-2 GMATCLUB MATH Q [#permalink]

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New post 15 Mar 2014, 02:47
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.

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Re: M13-2 GMATCLUB MATH Q [#permalink]

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New post 15 Mar 2014, 05:50
3
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.
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Re: M13-2 GMATCLUB MATH Q [#permalink]

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New post 15 Mar 2014, 05:59
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.
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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 08 Jun 2014, 00:17
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.

Answer: D.


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
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Re: How many roots does this equation have? ||x+2|-2|=2 [#permalink]

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New post 08 Jun 2014, 03:38
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.

Answer: D.


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.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: How many roots does the equation || x +2 | - 2 | = 2 have? [#permalink]

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New post 30 Jan 2015, 21:04
2
||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

ANSWER: D
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Re: How many roots does the equation || x +2 | - 2 | = 2 have? [#permalink]

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New post 20 Apr 2015, 09:01
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.

Answer: D.


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
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Re: How many roots does the equation || x +2 | - 2 | = 2 have? [#permalink]

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New post 08 Oct 2015, 10:42
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.

Final Answer:

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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] 15 Jan 2018, 07:17
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