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Last time I saw Matt Cutts speak he said every search result is personalized, logged in or not. They use a lot of different identifiers to serve results. So yes, computer to computer the results can and will vary.

This is a good reason putting too much emphasis on rankings can be problematic.

If you want to check rankings, I recommend using a rank checker tool like the one Aaron Wall offers... http://tools.seobook.com/firefox/rank-checker

Do you have a Google Places account? Is it their website or their Google Places listing that is ranking for the keyword? Do you have "Web Design Memphis" or a similar phrase on your homepage title tag? Do you use that phrase on your site?

I've found including all of the geo information in the footer generally works well. Creating a link between your Google Places account and your web page could help as well.

Yes a well optimized Google Places page can have a huge impact on getting local query traffic.

Hope you get it worked out. Good luck!