Great! You're making progress
So, you can add a definition query to your precinct layers to show only those precincts you want.
- Right-click the layer in the table of contents and click Properties.
- Click the Definition Query tab on the Layer Properties dialog box.
- Click the Query Builder button.
- Build a query expression. In your case, this will be something like:
"Field name containing attributes that define the precincts you want" in (list, of, attributes, separated, by, commas)
The query builder makes it easy because you can just double click the field you want from the list, and then get the unique attribute list and double click the values that you want to have in the list. The "in" statement makes it a list of values.
This should limit the display of the layer to just those precincts you choose.
Here is the help file on definition queries, if you need more detail:
ArcGIS Help 10.1