I've read adding resazurin (Alamar blue) or a tetrazolium salt to the tubes would give a change in colour in case of bacterial growth. If this is the case, would this method tell me the MIC or the MBC?
If your compound precipitates or is insoluble at higher concentrations, you won't get an accurate MIC value. When I work with compounds that have a poor solubility in water, I make a highly concentrated solution in DMSO and dilute this into broth to make my starting concentration. Then do serial dilutions as you would normally. You need to do a control with that DMSO concentration in broth to make sure it's not affecting your cells. 3% and lower is fine but above this you start to see decreased cell growth relative to no DMSO.