It is currently 18 Feb 2018, 18:10

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 43792
What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

### Show Tags

29 Aug 2017, 01:02
2
KUDOS
Expert's post
8
This post was
BOOKMARKED
00:00

Difficulty:

65% (hard)

Question Stats:

64% (01:11) correct 36% (01:13) wrong based on 202 sessions

### HideShow timer Statistics

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
[Reveal] Spoiler: OA

_________________
Director
Joined: 18 Aug 2016
Posts: 628
Concentration: Strategy, Technology
GMAT 1: 630 Q47 V29
GMAT 2: 740 Q51 V38
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

### Show Tags

29 Aug 2017, 01:44
1
This post was
BOOKMARKED
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
_________________

We must try to achieve the best within us

Thanks
Luckisnoexcuse

VP
Joined: 26 Mar 2013
Posts: 1427
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

### Show Tags

30 Aug 2017, 02:52
4
KUDOS
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

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

BSchool Forum Moderator
Joined: 26 Feb 2016
Posts: 2039
Location: India
GPA: 3.12
What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

### Show Tags

31 Aug 2017, 19:22
1
KUDOS
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)
_________________

Stay hungry, Stay foolish

Manager
Joined: 07 Jun 2017
Posts: 176
Location: India
Concentration: Technology, General Management
GMAT 1: 660 Q46 V38
GPA: 3.6
WE: Information Technology (Computer Software)
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

### Show Tags

31 Aug 2017, 21:20
1
This post was
BOOKMARKED
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
_________________

Regards,
Naveen
email: nkmungila@gmail.com
Please press kudos if you like this post

Intern
Joined: 06 Apr 2016
Posts: 20
Re: What is the range of the roots of ||x – 1| – 2| = 1? [#permalink]

### Show Tags

01 Sep 2017, 23:34
1
This post was
BOOKMARKED
||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
Display posts from previous: Sort by