Thanks! In this particular problem, x is an integer that can only take on values [1,2,...,+infinity] so I'll try to use Reduce instead, with a specification for allowed values of x.
I find that using Reduce and adding a stipulation that the variable be real valued helps here.
Reduce
In[312]:= Reduce[(5/6)^x <= 1/20 && Element[x, Reals], x] (* Out[312]= x >= (2 Log[2] + Log[5])/(Log[2] + Log[3] - Log[5]) *)
