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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: 50658
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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29 Aug 2017, 01:02
3
12
00:00

Difficulty:

55% (hard)

Question Stats:

59% (01:46) correct 41% (01:44) wrong based on 351 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: 26 Mar 2013
Posts: 1882
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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30 Aug 2017, 02:52
8
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: 18 Aug 2016
Posts: 625
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]

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29 Aug 2017, 01:44
3
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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What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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31 Aug 2017, 19:22
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)
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Joined: 07 Jun 2017
Posts: 174
Location: India
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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, 21:20
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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Please press kudos if you like this post

Intern
Joined: 06 Apr 2016
Posts: 26
GMAT 1: 720 Q49 V40
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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01 Sep 2017, 23:34
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: 30
GPA: 4
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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17 Sep 2018, 02:54
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
What is the range of the roots of ||x – 1| – 2| = 1? &nbs [#permalink] 17 Sep 2018, 02:54
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