Alternate and alternative

I'd say (as you probably realize) that "alternate" is more popular as an adjective and "alternative" is more popular as a noun.

I took the alternate route.

I use aspirin as an alternative to ibuprofen.

I realize this doesn't answer your question.

They can be used interchangeably in many situations.

alternative suggests to me the free choice a little more -- an alternative method -- if you want a different way of doing it, you can choose this one.

alternate suggests to me a little more the idea of what's available -- an alternate road -- there are two (or more) ways available for getting somewhere.

OseeCould you tell me what's the difference between the two words as adjective?

This thread may be of some help:

Good point, Ray. "Alternate Tuesdays" is an adjective. There's also "alternating shifts."

Thank you all guys very much. So, by my understanding, I should say "an alternative method to solve this problem is this ...." Is this right?

Osee I should say "an alternative method to solve this problem is this .." Is this right?

That would be an excellent choice.