Hi Stan -
Remember that every item in every category must be used once and only once for each of the people/places/things in the puzzle. Therefore Art must have a room #, and Pre-Calculus must have a room #. Clue #8 says that one of those two (either Art or Pre-Calc) is in room #114, and the other is Avery (who you already know isn't room #129). So any way you slice it, Art is either in room #114 or in Avery's room (which definitely isn't 129). The same also goes to Pre-Calc. It is impossible for either Art or Pre-Calc to be in room #129.
When you see a clue like that - i.e. Of A or B, one is C and the other is D - always be on the lookout for data already on the grid that tells you what C can or cannot be in relation to D's category, and vice versa. You can often swap this data into the original clue to make additional TRUE/FALSE conclusions.