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Could you tell me what's the difference between the two words as adjective? Thanks.
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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.

Either way, both choice and availability enter into it, so the subtle difference may be entirely imaginary on my part.

See the dictionary for other meanings that are not synonymous.

CJ
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OseeCould you tell me what's the difference between the two words as adjective?
This thread may be of some help:
http://www.EnglishForward.com/English/AlternativeSaturdayAlternateSaturday/wkxwm/post.htm#721530
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?
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Osee I should say "an alternative method to solve this problem is this .." Is this right?
That would be an excellent choice.