GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 23 Oct 2019, 00:42

### 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

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58438
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

29 Aug 2017, 02:02
3
16
00:00

Difficulty:

55% (hard)

Question Stats:

62% (01:49) correct 38% (01:40) wrong based on 404 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

_________________
SVP
Joined: 26 Mar 2013
Posts: 2341
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

30 Aug 2017, 03:52
9
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

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

##### General Discussion
Current Student
Joined: 18 Aug 2016
Posts: 602
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, 02:44
4
2
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
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3333
Location: India
GPA: 3.12
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

31 Aug 2017, 20:22
1
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)
_________________
You've got what it takes, but it will take everything you've got
Manager
Joined: 07 Jun 2017
Posts: 161
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, 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
_________________
Regards,
Naveen
email: nkmungila@gmail.com
Please press kudos if you like this post
Intern
Joined: 06 Apr 2016
Posts: 25
Location: India
Schools: Desautels '21
GMAT 1: 720 Q49 V40
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

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
Intern
Joined: 09 Apr 2018
Posts: 31
GPA: 4
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

17 Sep 2018, 03:54
1
I just applied a rule for complex absolute value equations:

If the equation contains 1 variable and 1 or more than 1 constant(s) in 1 or more than 1 absolute value expressions, the only two cases we have to consider are (1) that the expressions have the same sign and (2) that they have different signs.

Applied to the question here I computed the following:

case (1):

(x-1)-2=1
x-3=1
x=4

case (2)
-(x-1)-2=1
-x+1-2=1
-x-1=1
-2=x

Since we are asked for the range (= highest value-lowest value), we have to perform this final step to get to the answer: 4-(-2)=6.

Please hit Kudos if you liked this approach
Math Expert
Joined: 02 Sep 2009
Posts: 58438
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

24 Dec 2018, 04:53
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

_________________
SVP
Joined: 03 Jun 2019
Posts: 1746
Location: India
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

Updated on: 08 Oct 2019, 04:00
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

Case 1: |x-1|-2 = 1
|x-1| = 3
x = 4 or -2

Case 2: |x-1|-2 = -1
|x-1| = 1
x = 2 or 0

Set of roots x = {-2,0,2,4}
Range of roots = 4 - (-2) = 6

IMO D
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com

Originally posted by Kinshook on 08 Oct 2019, 03:13.
Last edited by Kinshook on 08 Oct 2019, 04:00, edited 1 time in total.
Manager
Joined: 10 Dec 2017
Posts: 85
Location: India
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

08 Oct 2019, 03:31
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

lx-1l=y
ly-2l=1
y-2=1 or y-2=-1
y=3 or y=1
lx-1l=3 or lx-1l=1
x-1=3 or x-1=-3 or x-1=1 or x-1=-1
x=4 or x=-2 or x=2 or x=0
x=-2 smallest
x=4 largest
range=4-(-2)
D:)
Director
Joined: 19 Oct 2018
Posts: 995
Location: India
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

### Show Tags

08 Oct 2019, 03:56
Slope of each line is 1 or -1

Hence the range of the roots of $$||x – 1| – 2| = 1$$= 1+1+1+1+1+1=6 (clearly seen from the graph)

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

Attachments

Untitled.png [ 4.78 KiB | Viewed 368 times ]

What is the range of the roots of ||x – 1| – 2| = 1?   [#permalink] 08 Oct 2019, 03:56
Display posts from previous: Sort by