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Math Expert V
Joined: 02 Sep 2009
Posts: 60559
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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4
21 00:00

Difficulty:   55% (hard)

Question Stats: 62% (01:49) correct 38% (01:41) wrong based on 471 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: 2344
Concentration: Operations, Strategy
Schools: Erasmus '21 (M\$)
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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10
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 P
Joined: 18 Aug 2016
Posts: 592
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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4
3
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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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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Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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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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GMAT 1: 720 Q49 V40 Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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

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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 V
Joined: 02 Sep 2009
Posts: 60559
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

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SVP  D
Joined: 03 Jun 2019
Posts: 1940
Location: India
GMAT 1: 690 Q50 V34 What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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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

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  G
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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

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:)
VP  V
Joined: 19 Oct 2018
Posts: 1293
Location: India
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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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 1034 times ]

VP  P
Joined: 24 Nov 2016
Posts: 1086
Location: United States
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|≥0 when x≥1 (positive); |(x-1)-2|=1…|x-3|=1;
|x-3|≥0 when x≥3: (x-3)=1…x=4=valid (x≥3)
|x-3|<0 when x<3: -(x-3)=1…-x+3=1…x=2=valid (x<3)

|x-1|<0 when x≤1 (negative); |-(x-1)-2|=1…|-x-1|=1
|-x-1|≥0 when x≤-1 (positive): (-x-1)=1…-x=2…x=-2=valid (x≤-1)
|-x-1|<0 when x>-1 (negative): -(-x-1)=1…x+1=1…x=0=valid (x>-1)

valid solutions: x={4,2,-2,0} range=largest-smallest=4-(-2)=6

Ans (D) Re: What is the range of the roots of ||x – 1| – 2| = 1?   [#permalink] 11 Nov 2019, 08:51
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