Do you need a router for XBOX live if you have a wireless adapter?

when you get a wireless modem, and you want to play your 360 on it do a few things. first check and see if your connection is open or closed. if it is open, then don't worry. but if it is closed your going to have to call your internet provider and get the passWord for the modem. then you need to. buy a Xbox 360 wireless router. it costs around $100. then just unplug your Ethernet cable, and ug in the wireless adapter to the USB port in the back of the xbox. then go to the xbox settings for xbox live and there you go.