Learn Arabic (Modern Standard): المطعم في الفندق.

In Arabic, there is no direct equivalent for the present tense word 'is'. To say 'The restaurant is in the hotel', you simply place the subject and the location next to each other: المطعم في الفندق (literally: 'The restaurant in the hotel').

Since this lesson is about attached possessive pronouns, how would I change this to say 'My restaurant is in his hotel'?

You drop the definite article الـ (al-) from both words and add the attached pronouns to the end. 'My restaurant' becomes مطعمي (maT'am-ii) and 'his hotel' becomes فندقه (funduq-hu). The new sentence is مطعمي في فندقه.

How do I pronounce في الفندق together? Does the vowel sound change?

Yes, they blend together! When the preposition في (fii) is followed by a word starting with the definite article الـ (al-), the long 'ee' sound and the 'a' in 'al' are dropped so the words flow smoothly. You pronounce it as 'fil-funduq' instead of 'fii al-funduq'.

Does the word المطعم (the restaurant) share a root with other food words?

Exactly. Most Arabic words are built from three-letter roots. The root for eating or food is ط ع م (T-'-m). Adding the 'ma-' prefix often indicates a 'place where an action happens', turning it into مطعم (maT'am), meaning a place of eating.