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When you analyze a problem you look it over which is what analyzing means. You look over the problem and then you solve it. When you solve a problem you solve it and you use certain steps and solve it but of course everyone has there ways to solve a problem but some people have ways to solve it by just analysing it. That is the difference.
It depends on the problem: you may have to use integer programming rather than linear programming.
There is no such thing as "solving integers". You can solve an equation, which means finding all the unknowns in that equation, but you can't solve an integer.
NO!
I don't have a single clue... go on wikipedia to find out!