Getting breakouts around my mouth because of hair removal cream?

i use nair on my upper lip, and a couple of days later i get little breakouts. i know its the hair follicles trying to break through, but what can i do about it? any suggestions on a good cleanser? i have dry skin, i use lemon juice every night and wash it off with water in the morning because it works for me, and i don't get acne or any other breakouts. I don't want to stop removing the hair from around my mouth because even though i have light coloured hair, you can still see it when i wear make up. thanks! :)