Question 179445
You made the mistake.
this is the problem |x-1/3|=|1/2x+1/6| 
x-1/3=1/2x+1/6  ------>This is wrong.
You can not just pull it out from the absolute
The theory is   If |A|=|B|  Then ---> A^2 = B^2
|x-1/3|=|1/2x+1/6|  -----> (x-1/3)^2 = (1/2x+1/6)^2
(x-1/3)^2 = (1/2x+1/6)^2
x^2 -(2/3)x + (1/9)  = (1/4)x^2 +(1/6)x + (1/36)-------(1)
36 * Eq(1)
36x^2 - 24x + 4 = 9x^2 + 6x + 1
27x^2 - 30x + 3 = 0
divide by 3
9x^2 - 10x + 1 = 0
(9x -1 )(x-1) = 0
Then x = 1/9  ,  1  Ans.