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

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Math Expert
Joined: 02 Sep 2009
Posts: 46280
What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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29 Aug 2017, 02:02
2
9
00:00

Difficulty:

55% (hard)

Question Stats:

64% (01:12) correct 36% (01:13) wrong based on 227 sessions

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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

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Joined: 18 Aug 2016
Posts: 634
GMAT 1: 630 Q47 V29
GMAT 2: 740 Q51 V38
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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29 Aug 2017, 02:44
1
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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Joined: 26 Mar 2013
Posts: 1677
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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30 Aug 2017, 03:52
5
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

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

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Joined: 26 Feb 2016
Posts: 2834
Location: India
GPA: 3.12
What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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31 Aug 2017, 20:22
1
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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Manager
Joined: 07 Jun 2017
Posts: 174
Location: India
Concentration: Technology, General Management
GMAT 1: 660 Q46 V38
GPA: 3.6
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Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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31 Aug 2017, 22:20
1
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
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Intern
Joined: 06 Apr 2016
Posts: 23
GMAT 1: 720 Q49 V40
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

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02 Sep 2017, 00:34
2
||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] 02 Sep 2017, 00:34
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