I have a double-basin kitchen sink, and the drain line from the garbage disposal is completely clogged. I've been unable to clear it with drain opener, so I'm looking at replacing the P-trap, but space is a bit tight. The pipe nut just on the drain side of the P-trap won't come loose – I've tried WD-40 and several different wrenches. Given the pictures that follow, how much of the drain system should I try to replace? Or should I simply call a professional, given the tight spacing between couplings?
The non-disposal side of the sink is not clogged, and is draining normally. I'd prefer to not replace that part of the drain system if I don't have to, but I'm willing to if I can do the replacement myself and need to replace the whole system to do it sanely.
EDIT:
I'm pretty sure I'm turning it the right direction – it won't budge in either direction, though. I decided to punt and cut the downpipe coming from the disposal leading to the P-trap (intending to snake the trap and replace the downpipe with an extended with two slip joints) – and yes there is a slip joint from the disposal to the downpipe – and the clog is in the disposal itself. The pipes are completely clear. Since the disposal has been giving me problems (the motor won't turn the blades when I turn power on) for about 4 months, it's time to just replace the disposal. I'm not comfortable doing that myself, so it's time to spend some money on a professional. Thanks for all the help.
Image 1: the P-trap below the disposal: 
Image 2: the joint for both "halves" of the sink drain: 
The longer pipe coming "down" comes from the P-trap below the non-disposal side of the sink.
Image 3: P-trap below non-disposal side of the sink: 
Image 4: the non-disposal side where the dishwasher waste line comes in: 
Remember, hand tight is not enough. Use adjustable pliers to tighten (but not over tighten) all slip-joint fittings.
Best Answer
I disagree with BMitch as far as if the P-trap will come apart. (Possibly the first time we've disagreed?) The metal nut could not be where it is if the joint did not separate. The rest of his answer I agree is the easiest fix. However, if possible, physically cleaning the trap is preferable to a chemical approach. I'm assuming there is a slip fitting just out of the picture, just as there is in non-disposal side. By detaching the metal ring in the first photo and the unseen slip joint, the trap should be able to be removed.
Even without a slip joint, the pipe could be detached right at the disposal itself. This could actually be preferable as it would allow the P-trap to swivel, which may help in loosening the nut. Are you sure you are trying to turn the nut the correct direction? I'm not saying you don't know how to loosen fasteners. The direction depends on which piece the nut is threaded onto, which is not always clear in plumbing.
The curved shoulder of the nut tells me it is threaded onto the U-bend, so the nut is turned anti-clockwise when looking down at it. It's hard to tell by photos, but you appear to have decent access to the nut from the left side as it is below the pipe coming in from the other side. Even though the opposite trap may be in the way for fully turning a wrench, you seem to have enough room to place the wrench handle near the trap and pull right.
As for counter force, if pulling on the pipe itself isn't working, try a large screwdriver wedged in the inside of the U-bend. If you have the right wrench, the ring nut shouldn't offer a lot of resistance.