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

Question

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

A. 0
B. 2
C. 4
D. 6
E. 8

M37-17

Bunuel · Accepted Answer

Official Solution:

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 
 
CASE 1:
 
 |x – 1| – 2 = 1  
 
 |x – 1| = 3  
 
 x=4  or  x=-2 .
 
CASE 2:
 
 |x – 1| – 2 = -1  
 
 |x – 1| = 1  
 
 x=2  or  x=0 .
 
So,  x  can be -2, 0, 2, and 4. The range of values  = 4-(-2)=6 .

Answer: D

Luckisnoexcuse · Answer

||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

Mo2men · Answer

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

Answer: D

pushpitkc · Answer

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)

nkmungila · Answer

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
Answer D

Darknight2 · Answer

||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

laulau · Answer

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

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

Kinshook · Answer

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

satya2029 · Answer

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:)

nick1816 · Answer

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)

exc4libur · Answer

|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)

kaladin123 · Answer

Alternative Approach If a & b are two numbers on the number line, |a-b| represents the absolute value of the difference between them. Likewise, we can draw a couple of number lines to infer that x=\{0,2,-2,4\} nl3.jpg Hence, Range=Max-Min=4-(-2)=6

Crytiocanalyst · Answer

– 1| – 2| = 1
|x – 1| – 2 = 1 or |x – 1| – 2 = -1

|x – 1| – 2 = 1
|x – 1| = 3
x – 1 = 3 or x – 1 = -3
Therefore x = 4, -2

if |x – 1| – 2 = -1
|x – 1| = 1
x – 1 = 1 or x – 1 = -1
 x = 2, 0

range equals 4+2 = 6

Therefore IMO D

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

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