Around town

la mairie is where le maire works

You met le maire at L185. la mairie is his building — the town hall, and in France the place you go for a birth certificate, a marriage, or a passport form. In a large city it may also be called l'hôtel de ville, which has nothing to do with hotels.

The mayor works at the town hall.
Le maire travaille à la mairie.

Three words for where people live

le centre-villethe town centrecarded at L36
la banlieuethe suburbsthe ring around a city
la capitalethe capitalthe country's main city

la banlieue is singular in French where English says the suburbs: Ils habitent en banlieue. Note the preposition — en banlieue, with no article, exactly like en ville.

gratuit

gratuit means free of charge, never free as in unconstrained — that is libre, which you have from L9. The two are not interchangeable:

Le musée est gratuit.The museum costs nothing.
La place est libre.The seat is unoccupied.

Its feminine adds a -e and changes the sound: gratuit / gratuite, where the final t is pronounced.

The capital's museum is free.
Le musée de la capitale est gratuit.

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