NSolve[((900/454) (1816000 E^(-200 t/681) (-1 + E^(1782 t/6583)))/2673) == 50, t, Reals]
(*{{t -> 0.1402163037}, {t -> 143.2657137}}*)
Took about 40 seconds on my PC.
For inequality, one is supposed to use Reduce[]. But it was taking long time on my end. You can also use FindInstance if you just want one solution:
FindInstance[((900/454) (1816000 E^(-200 t/681) (-1 + E^(1782 t/6583)))/2673) <= 50, t]
(* {{t -> 0}} *)
