I would like to ask everyone if it is appropriate to post a solution to a "problem" in my web development projects.
It would be in a format such as "Needed a way to solve (insert problem here) and here is the solution that I came up with".
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Just ask the question on how to do it then post your answer. This is a specifically allowed way of doing it.
It makes more sense to ask "How do I combine tables and CSS?" than to state "Here's how I combined tables and CSS".
Note that you won't get points for accepting your own answer nor will it rise to the top (unless it gets the most votes). If you want to be sneaky, don't ask the question until you have the answer ready to post in. That makes it more likely that people will vote you up (if you're that way inclined).
Like Pax said, you can answer your own question.
Another option is to create a community wiki. I think the community wiki "questions" are great for solutions to potential questions and/or some how tos.
If your how to gets into article territory, then I think a blog posting is probably the most appropriate. That line isn't well defined though IMHO.
If you search StackOverflow first and can't find a similar/identical question already, AND you believe it was a non-obvious answer that other people will benefit from, then I can't see why not.
In this case you may want to create a normal question (phrase the title as a question) and immediately post your answer, as an actual answer. That allows other users to either indicate your answer is useful/correct or add their own answers.