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What is the range of the roots of ||x – 1| – 2| = 1?

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What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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New post 29 Aug 2017, 01:02
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Question Stats:

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Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:



What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


||x – 1| – 2| = 1
|x – 1| – 2 = 1 or |x – 1| – 2 = -1

if |x – 1| – 2 = 1
|x – 1| = 3
x – 1 = 3 or x – 1 = -3
giving us x = 4, -2

if |x – 1| – 2 = -1
|x – 1| = 1
x – 1 = 1 or x – 1 = -1
giving us x = 2, 0

now we have x=-2,0,2,4
range will be 4+2 = 6
D
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Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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New post 30 Aug 2017, 02:52
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Bunuel wrote:
What is the range of the roots of \(||x – 1| – 2| = 1\)?

A. 0
B. 2
C. 4
D. 6
E. 8


Another approach

Let \(z = x -1\)

\(||z| – 2| = 1\).............square both sides

\(z^2-4|z|+4=1\)

\(z^2-4|z|+3=0\)

\((|z|-3)(|z|-1)=0\)

\(|z|=3\) or \(|z|=1\)

substitute z from above

\(x–1=3\) or \(x–1=-3\)
\(x=4\) or \(x=-2\)

OR

\(x–1=1\) or \(x–1=-1\)
\(x=2\) or \(x=0\)

Range = Max value - min value

\(R=4-(-2)=4+2=6\)

Answer: D
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What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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New post 31 Aug 2017, 19:22
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The expression is in the form ||a| - 2| = 1

There are 4 possibilities, when trying to solve this equation
|-a -2| = 1
There are 2 options:
-(-a -2) = 1 => a +2 = 1 => a = -1
(-a - 2) = 1 => -a = 3 => a = -3

|a - 2| = 1
There are 2 options:
-(a - 2) = 1 => -a + 2 = 1 => a = 1
(a - 2) = 1 => a = 3

Hence, there are 4 values for a which are -3,-1,1,3

Coming back to the question, if we substitute a to be x-1

x - 1 = -3 => x = -2
x - 1 = -1 => x = 0
x - 1 = 1 => x = 2
x - 1 = 3 => x = 4

Hence, the range of the values is 6(Option D)
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Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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New post 31 Aug 2017, 21:20
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the values are
X-3=1
-X+3 = 1
-X-1 = 1
X+1 = 1

the values X are { 4, 2, 0, -2}

hence range = 4-(-2)
6
Answer D
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Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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New post 01 Sep 2017, 23:34
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||x-1|-2|=1

Removing the outer modulus, we get
|x-1|-2 = 1 (or) -1

|x-1|= 3 (or) 1

Now, if we remove the modulus for |x-1| we will get the following four possible values for x

x-1= 3 (or) -3 => x= 4 (or) -2
X-1= 1 (or) -1 => x= 2 (or) 0

Hence the four possible values of x are (-2,0,2,4)
=> Range= 4- (-2) = 6

Ans-> Option D
Re: What is the range of the roots of ||x – 1| – 2| = 1?   [#permalink] 01 Sep 2017, 23:34
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