Ceiling drywall edge support question

Hello,

I removed a soffit above a stand up shower in my bathroom. The replacement piece of drywall is 3'x4'. 2 sides of the drywall do not have joists above where the edges go. The builders put a 1x piece of wood on the top of one of the walls so you can screw the edges of the drywall into it, but the other side does not have that. On the side with the missing support the drywall edge will be unsupported about 2.75". There is a joist 2.75" away from that wall. The whole rest of the bathroom ceiling on that edge appears to be unsupported for 2.75" (I am only replacing one small section). I pushed on the edge of the existing ceiling drywall and it feels pretty firm even though not supported for 2.75" on the edge. Do I need to do anything about it? I cannot easily add blocking there because the builders put a 1x piece of wood on that same wall which goes to the OTHER room ceiling drywall.

2) Other question is the joists are 16 on center so two sides of the drywall will have screws every 16" at the edges. Is that good enough or do I need to add some 1x or something to have more frequent screws.

Thanks!

 

This is a fairly large patch, but still minor IMO.

You are OK on the side with the 1 x backer.
For the unsupported side, it's probably OK with just tape and mud (especially setting type compound).

If there's no room for a 1 x backer, I use the clips below. You only need a few and they work very good. Most any home store sells them.
The little tabs on the outside of the wall will be broken off after the new piece is inserted.

 

I agree with Brian, that it's fine.