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

Question

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

M13-02

Bunuel · Accepted Answer

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
 
Given:  ||x +2|-2|=2 . Two cases are possible:

1.  |x +2|=0 . In this case  x=-2 ;
 
2.  |x +2|=4 . In this case  x=2  or  x=-6 .

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

sondenso · Answer

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

imhimanshu · Answer

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

Thanks
H

imhimanshu · Answer

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

Bunuel · Answer

Possible? Yes. Good approach? No. As for the graph itself: graph.png Hope it helps.

WoundedTiger · Answer

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.

Bunuel · Answer

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)^4-16(x+2)^2=0 ;

((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.

thorinoakenshield · Answer

||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

EMPOWERgmatRichC · Answer

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:  D

GMAT assassins aren't born, they're made, 
Rich

Kinshook · Answer

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

Case 1: |x+ 2| - 2 = 2
|x+2| = 4
x = 2 or -6

Case 2: |x+ 2| - 2 = -2
|x+2| = 0
x = -2

x = {-6,-2,2}; 3 roots

IMO D

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

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