Tourbleshooting an Electrical Circuit?

My TV room was a garage before. One day several outlets in the room quit working, but the breaker did not throw. Suddenly a lamp still plugged into an inoperative outlet came on, but the light was very dim. I plugged something into another inoperative outlet and the light got so dim it just about went off. Then in just a few seconds the outlets quit again. I have attached a picture of the outlets that work and don't work.