Your sentence is grammatically correct, but is also a somewhat political statement.
Many people do not like being grouped together and labeled as adjectives, particularly when the adjective is a pejorative one: poor, impoverished, destitute, homeless, etc.
It is more politically correct to say "Many of the city's lower-class residents are unable to find housing." Or, "Many of the city's residents who live in poverty are unable to find housing."
The use of "residents" here as the noun to be qualified by an adjective or by an epithet or other descriptor gives the people behind the name dignity and humanity.
But you will often see sentences like your original one in older texts, and certainly many people today still tend to speak that way. And like I said, the grammar is fine.