In Joseph Gallian's book Contemporary Abstract Algebra, he calls the above the "Socks-Shoes Property" for any group elements x and y.
Exercise 1: Put your socks on followed by your shoes. Now see if you can remove your socks without removing your shoes (or cutting your socks). If you can, then your shoes are too big!!
Exercise 2: Put your shoes on followed by your socks... now go to work/school like that. If no one says anything to you about it, then you must be hanging out with too many mathematicians!