Learn English – How is double negation interpreted in English

It might be easier to understand if we use a syntactically simpler action, initially without negation...

1: You must eat (it is necessary that you eat)
2: You need to eat (note that to is grammatically required here)

For most purposes, #1 and #2 are semantically equivalent (it is necessary that you eat). But things change if we introduce negation. There's only one valid way to negate #1...

3: You must not eat (it is necessary that you do not eat)

But there are two ways to negate #2, and they have completely different meanings...

4: You need to not eat
5: You need not eat

In #4 the negation attaches to eat (i.e. - to the activity of eating), so it just means the same as #3 above.
But in #5, the negation attaches to need (i.e. - it's not actually necessary that you do not eat). You can either eat or not eat - it's a matter of personal choice, not constrained by any obligatory need.

Note that although it's perfectly grammatical, and not exactly an "unusual" form, #4 is a far less likely construction than #5. Probably in order to avoid any possible confusion, native speakers tend to use must rather than need in such constructions (or introduce "Do-support" per @hellion's comment below).

Inactive is better. This is not a grammar point, but a matter of style.

Using "not" introduces an ambiguity, as it could be taken to mean "... will not remain active", a"different meaning entirely. In such a sentence using "remain not [adjective]" is rarely the best way to say something.

Say "remain inactive" instead of "remain not active". Say "remain unhappy" instead of remain not happy". If the adjective doesn't have a negation you may rephrase: Instead of "remain not red" say "still hasn't turned red" (or something similar depending on the exact meaning)