Adequan is administered via an intramuscular (IM) injection. This means you will use the needle and syringe and poke it into the largest muscle group nearest the affected joint to ensure that it gets to work on the right area.
This means if your dog is having trouble with a rear leg, you'd go in the large muscle group in the thigh area. The vet tech coming out to your home can show you the proper way to do it to avoid hitting anything dangerous (such as the sciatic nerve).
A word of warning however...IM injections do hurt quite a bit more than something like a subcutaneous injection, so giving the injection yourself may not be easy, depending on the disposition of the dog. Asking the tech coming out to show you how to do it yourself and then evaluating how the dog reacts will be the real test of whether this is something you should attempt at home.
I hope this helps.