jmc

First we divide by 3, writing the right side as a mixed number: $$x-5\le 2\frac{1}{3}.$$ If we add 5 to both sides we get $$x\le 7\frac{1}{3}.$$ The positive integers solving this are $1,2,3,4,5,6,7$. There are $\boxed{7}$ integers on this list.