Don't go toward the low end of your budget. $40 a night in Nashville will be gross and unsafe.
You didn't give any dates, so I just entered mid-September and came up with some great bargains at really nice hotels. I use www.tripadvisor.com to search hotels because it includes reviews from travelers. Use the arrow at the far left to adjust the price range.
If you will add your dates, I'll give you some specific hotel recommendations. In general, the area near the airport is less expensive than downtown or West End.
EDIT: Thanks for adding the date. In looking at availability for that date, I would suggest the following with have prices right at $80 (plus tax). Some of these rates are pre-paid, non-refundable rates. To book them, I suggest going to the Convention and Visitors Bureau site at www.visitmusiccity.com, click on Lodging at the bottom, and enter your dates. The prices are the same at the other major sites, but I feel confident the CVB is watching over you.
Doubletree Guest Suites Nashville Airport -- $79.90
2424 Atrium Way
Radisson Hotel Nashville Airport -- $77
1112 Airport Center Dr.
Holiday Inn Opryland Airport --$81.95
2200 Elm Hill Pike
As I said before, the really low price hotels scare me and, I believe, should be avoided.
former travel agent