Here is the first mistake: You didn't use brackets! Actually you mean:

1/3 - (x+1)/5 = x/3

Better learn some LaTeX.To multiply through by 15 is the right idea:

15(1/3-x/5+1/5) = 15(x/3)

15/3-15x/5+15/5 = 15x/3

5-3x+3=5x

8-3x=5x

8/8=8x/8

x=1

The book gets 1/4

have I made a mistep, Ive done this problem 3 times?

Expand the bracket, collect like terms - and you'll get the given result.