How to find the maximum value of an integer that satisfies some inequality

Your approach to solve for general j using Reduce is correct. However, you can also use Maximize or NMaximize as:

Maximize[{j, 2^j/(j + 1) <= 10}, j, Integers]
(* {6, {j -> 6}} *)

Or even more compactly, as JM notes:

ArgMax[{j, 2^j/(j + 1) <= 10}, j, Integers]
(* 6 *)

If you want to use the results from Reduce, you could do:

Max[j /. Solve@ Reduce[2^j/(j + 1) <= 10, j, Integers]]