Danielle Thurston, 38, was last seen at the home in the 17100 block of Sunderland at 5:30 p.m. Monday. She left on foot and never came back.
Police said it was her first day as a resident at the home. Thurston is in good physical condition but has a mental disorder, police said.
Thurston is 5 feet, 3 inches tall, and weighs 150 pounds. She has light blonde hair and blue eyes. She was last seen wearing a flower shirt, blue jeans, and white shoes.
Anyone with information is about Thurston's whereabouts is asked to call police at 313-596-5840 or Crime Stoppers at 1-800-SPEAK-UP.