I don't really understand the basic principles behind this and so I am looking for a good explanation on it.
My lecture notes use inequalities like $dS \geq 0$, yet in a question involving calculating the entropy change when compressing Argon, the total entropy change was negative.
Then there is the notion entropy change of the system and entropy change of the surroundings? Can any of these be negative?
In other words, when can entropy be negative? In what circumstances is it strictly positive?