I am looking for the domain.

Question

f(x)=log(sub3)(5x+1)
5x+1<=0
(5x/5)<=(-1/5)
x<=(-1/5)
x>(-1/5)
(-1/5,oo) -domain

Answer 1

5x +1 can range from 0+ to infinity. Logs of zero and negative numbers are not allowed, to any base.
5x can then range from just above -1 to infinity.
x can therefore range from (> but not = 0.2) to infinity. Your answer is correct

You seem to have the < direction wrong in your first 3 inequalities, but you reversed it at the end.