Learning about different problem solving strategies and when to use them will give you a good start. Problem solving is a process. Most strategies provide steps that help you identify the problem and choose the best solution. There are two basic types of strategies: algorithmic and heuristic.
It should be 'No vacancies' (plural). Vacant means empty/free. Vacancy is the state of being empty, therefore 'no vacancies' = no empty rooms.
Definition of inanimate. 1 : not animate:a : not endowed with life or spirit an inanimate objectb : lacking consciousness or power of motion an inanimate body. 2 : not animated or lively : dull.