Grade 12th passAlgebra
the number of solution of |x-1| ={x} where {} represents fractional function,
2 Answers
Deepak Kumar Shringi
Samyak Jain
Method 1 : Draw graphs of |x – 1| and {x} and note values of x where intersection of two graphs occur.
Method 2: By algebra.
|x – 1| = {x} …...............(1)
Case 1 : Let x 
1 & x 
Z 
{x} = 0
We can write (1) as x – 1 = 0 or x = 1
Taking intersection, we get x = 1 .…..........(2)
Case 2 : Let x > 1 & x 
Z
(1) becomes x – 1 = {x} or [x] + {x} – 1 = {x}
i.e., [x] = 1 1 
x
Taking intersection, we get 1
Case 3 : 0 x 
1 & x Z
So (1) is 1 – x = 0 i.e. x = 1
Taking intersection, we get x 
…............(4)
Case 4 : 0 Z
So (1) can be written as 1 – x = {x} or 1 – [x] – {x} = {x}
{x} = (1 – [x] ) / 2 We know that for 0 x 1, [x] = 0
Case 5 : –1 x 1 – x = [x} or [x] = 1 – 2{x}
For –1 x
So, 1 – 2{x} = –1 or {x} = 1, which is not true.
x …...........(6)
Similarly, for negative x there is no solution of (1).
Taking union of (2), (3), (4), (5) & (6), we get x 
[1,2) 
1/2 ,
which are the solutions of |x – 1| = {x}.
Hence, there are infinitely many solutions of |x – 1| = {x} .
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