How do I connect my new sink tailpiece to a glued-in PVC P-trap?

You're going to have to cut that glued trap out if you want a reliable seal. Since the trap is glued solid and doesn't line up with your new sink, there's no magic fitting that is going to bridge an offset like that while also being watertight. Trying to hunt down a specific nut for an old, misaligned adapter is ussually more trouble than it’s worth and often ends in a slow leak anyway. Take a hacksaw or a PVC cutter and snip the horizontal pipe coming from the wall, leaving about three inches of straight pipe sticking out. You need that room to glue on a **trap adapter**. This is a small fitting that converts your heavy schedule 40 PVC into a threaded connection that accepts standard 1.5-inch tubular plumbing. It’s the industry standard for under-sink drainage becuase it allows for future adjustments. Once that adapter is glued on, go buy a 1.5-inch **polypropylene P-trap** kit. These are the slip-joint style ones. They're much better for this situation because the trap itself can swing 360 degrees around the adapter and slide up and down on the tailpiece. That flexibility is exactly what you need to hit that one-inch offset without putting stress on the joints. If you try to force a hard-piped connection to meet, it’ll eventually crack or pull the sink flange loose. If the black tailpiece from the sink is still too short to reach the new trap, just grab a slip-joint extension tube. They come in various lengths and can be cut down easily. Just make sure you use the beveled plastic washers that come in the kit and don't over-tighten them. Hand tight plus a quarter turn with pliers is usually plenty. This setup makes it way easier to clean out the trap if you ever drop a ring down the drain later on.

So if I cut that pipe coming out of the wall, is there a risk of it being too short to glue the new adapter on? I'm super nervous about making permanent cuts like that. Also, I saw those 'flexible' accordion-style pipes at the store. Are those any good for this offset problem or are they just a disaster waiting to happen? I realy want to avoid messing with the wall part if I can, but the slip-joint trap sounds way easier than what I have now.

Whatever you do, stay far away from those accordion pipes. They’re basicly gunk collectors—all the soap scum and hair gets caught in those ridges and they start smelling within a month. Plus, they almost always leak at the seals beacuse they don't sit flush. Regarding the pipe coming from the wall, as long as you have about an inch and a half of clean pipe, a trap adapter will seat just fine. If you’re realy nervous about the length, look for a "deep socket" adapter for more surface area to glue. It’s better to do it right now with rigid PVC than to deal with a slow-draining, stinky sink for the next year.

If you're realy stressed about the pipe being too short, grab a 'socket saver' bit just in case. It lets you drill out the old pipe from inside a fitting if you mess up. But honestly, even if you only have an inch, you can usually make it work with a standard coupling. Also, when you do cut it, make sure you sand down the edges on the inside of the pipe. If there are any plastic burrs left over, they'll catch hair and cause a backup way faster than you'd think. It's a quick step that saves a lot of annoyance later on.

One thing to watch out for: check the diameter of your tailpiece. Bathroom sinks are often 1-1/4 inch, while kitchen sinks and most trap kits are 1-1/2. If they don't match, you'll need a specific reducing washer to bridge that gap properly.

When you're putting the slip joints together, make sure the tapered side of that plastic washer faces toward the joint, not the nut. I see people put them on backward all the time and then wonder why it drips. Also, don't use any pipe dope or tape on those slip threads; the washer does all the sealing. Hand-tightening is usually enought to prevent leaks without cracking the plastic.