On Mint version 16, in a BASH script - run using the live DVD:-
I create a directory /mnt/XXXXXXXX
mount /dev/sdax /mnt/XXXXXXXX
Do some processing (a backup actually).
I then want to delete /mnt/XXXXXXXX
But the directory is still in use for an use for an indefinite time.
Since data is written to these mount-points, I really need to umount them.
As long as any file in the mounted partition is open (that means file or directory), or as long as you have a terminal open to /mnt/sdax, or as long as you have a file manager open to /mnt/sdax, then the partition is in use and you can't unmount it.
When you are done processing files, cd out of the mounted partition, close the file manager or cd to a directory not in /mnt. then try to umount the partition.