We discuss an elegant method, introduced by Carlen and Loss, for finding extremizers of the Hardy-Littlewood-Sobolev inequality in certain cases. The talk will begin by introducing both the Hardy-Littlewood-Sobolev inequality and the notion of an extremizer. Briefly, we discuss Lieb's celebrated paper showing that extremizers for HLS exist. We will then discuss two symmetries of the problem: one which creates radial structure and one which conflicts with such structure. Finally, using the method of Carlen and Loss, we will show how these competing symmetries can be employed to construct an extremizer.