How many integers are there in the solution set of $|x-2| \leq 5.6$?
Since $|x-2|$ is an integer, it can equal 0, 1, 2, 3, 4, or 5.  If $|x-2| = 0$, we have only one solution for $x$.  Otherwise we have 2.  This leads to a total of $\boxed{11}$ integers in the solution set.