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M13-02

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Bunuel
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M13-02 [#permalink] New post   16 Sep 2014, 00:48
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45% (medium)

Question Stats:

59% (01:27) correct 41% (01:16) wrong based on 153 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
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D
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Bunuel
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Re M13-02 [#permalink] New post   16 Sep 2014, 00:48
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Official Solution:


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
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prashd
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Re: M13-02 [#permalink] New post   16 Nov 2014, 07:12
Hi,

Can you please explain how we get -6 as one of the root in the second case .
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Bunuel
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Re: M13-02 [#permalink] New post   16 Nov 2014, 07:31
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prashd wrote:
Hi,

Can you please explain how we get -6 as one of the root in the second case .


|x +2|=4. Two cases:

When x >= -2 we'd have x + 2 = 4 --> x = 2.

When x < -2 we'd have -(x + 2) = 4 --> x = -6.
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Re: M13-02 [#permalink] New post   26 Nov 2014, 15:14
why is I x +2 I = 4 an equation to find the second root?
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Re: M13-02 [#permalink] New post   27 Nov 2014, 03:34
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rsamant wrote:
why is I x +2 I = 4 an equation to find the second root?


\(||x +2|-2|=2\) --> denote |x+2| as y, so \(|y-2|=2\) --> y - 2 = 2 or -(y - 2) = 2 --> y = 4 or y = 0. Thus \(|x +2|=4\) or \(|x +2|=0\).
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Re: M13-02 [#permalink] New post   11 May 2017, 16:43
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Bunuel wrote:
How many roots does the equation \(|| x +2 | - 2 | = 2\) have?
A. 0
B. 1
C. 2
D. 3
E. 4


Anything inside an absolute value can be positive or negative. You have to test both. Since there are two absolute values here, there are four possibilities for a root. We need to test (inner)(outer) absolute values which will be ++, +-, -+, or --.

(++)
x+2-2 = 2
x = 2 one root

(+-)
-(x+2-2) = 2
-x-2+2 = 2
x = -2 second root

(-+)
-(x+2)-2 = 2
-x-2-2 = 2
x = -6 third root

(--)
-(-(x+2)-2) = 2
x+2+2=2
x=-2 this root was already found

Three roots, which are x={-6,-2,2}. Answer is D
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aswindvc
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Re: M13-02 [#permalink] New post   24 Aug 2020, 09:33
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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

Answer: 1) |x+2|−2 = 2 => |x+2|=4, we get two values of x i.e., x=2 and x=-6.
2) |x+2|−2 = -2 => |x+2|=0, we get only one value for x i.e., x=-2.
Total, we have 3 values for x. So, option (D) is correct.
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Re: M13-02 [#permalink] New post   17 May 2022, 23:26
I think this is a high-quality question and I agree with explanation. Its a very easy question if one does not get confused by the double mod signs. A qualitative question indeed.
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Bunuel
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Re: M13-02 [#permalink] New post   12 Aug 2023, 08:53
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re: M13-02 [#permalink]
12 Aug 2023, 08:53
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