1/(3x+1)+1/(3x-5)=1 can be written:
3x-5+3x+1=(3x+1)(3x-5) by multiplying through by the common denominator.
6x-4=9x²-12x-5,
9x²-18x=1,
x²-2x=1/9,
x²-2x+1=1+1/9,
(x-1)²=10/9,
x-1=±√10/3,
x=1+√10/3 or 1-√10/3.
This is approximately:
x=2.0541 or -0.0541.