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Ann Arbor is an anomaly in the state. For a city, AA doesn't have the problems that plague most other Michigan cities. In fact, most would say there are no bad abandoned or crime pockets within the city; that said, safety and crime should be little to no concern there.
Public schools I hear are great in Ann Arbor. The city has the two largest public high schools in Michigan, with a third opening soon. The two existing schools have strong ratings, creating symmetry in AA- i.e., there are 'well-to-do' families on both sides of the city. Some cities, for instance, have the token schools, rail road tracks or river that socially and economically separate the city.
Though the idea of Ann Arbor being a "College Town" is contested, I would argue that U of M, above anything else, has made the city a popular and desirable place to live. In my opinion, without the university, AA would either be another far-removed Detroit suburb searching for identity or a separate city with a distressed economy, e.g., near bye Bellville, Ypsilanti or Jackson. Furthermore, the university has made AA a 'collage' city, meaning, neighborhoods and business strips catered to college students and local residents weave together, creating interest and uniqueness.
Ann Arbor has many established neighborhoods on all sides of the city. If you look at a map of the city, established, architecturally unique neighborhoods are found within the island of highways, nearest to the city center. With your interests and price range, many would generally look just a mile or two out of central AA in the west or east-central parts of the city; though, virtually all sides have something to offer.
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