Final answer:
The Young Men's Christian Association (YMCA), influenced by the social gospel philosophy, built libraries and exercise rooms to help alleviate the challenges of urban life for the working class.
Step-by-step explanation:
The social welfare group that eased the problems of city life by building libraries and exercise rooms, among other community services, was the Young Men's Christian Association (YMCA).
Inspired by the teachings of Reverend Washington Gladden who advocated for the social gospel, organizations like the YMCA and the Salvation Army expanded their reach in American cities during the late 19th and early 20th centuries. These organizations pursued their mission of societal betterment by including amenities such as gymnasiums and libraries, as well as providing evening classes on health and hygiene to support the urban poor.