My problem is, the negative charges are not labeled, so I don't know when to use SN2 or E2. How can I determine this?
Reagents with a charge will often be provided with a counterion. For example, methoxide (MeO—) is often given as sodium methoxide (MeONa). We know that the sodium and oxygen have to be an ionic bond and that sodium has a positive charge, so the oxygen must have a negative charge.
The strength of the nucleophile/base is determined by the charge, and you can use that information to figure out what mechanism and product you will get by following the Big Daddy Flowchart, which you can find here: