Is the order placement of a city name in title tag very important?
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Is "Austin Plumbers" much different than "Plumbers Austin" in the eyes of the search engine and best practices?
I would think that Plumbers Austin would have more search volume. But Austin Plumbers is easier to work in to sentences in the body.
Does Google sees them as the same?
Is it ok to use Plumbers Austin in the title tag and use Austin Plumbers in the body or should I choose one target phrase and stick with just that? If so should the city name come first or last?
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According to Google's Adwords Keyword Tool, "austin plumbers" gets more search volume (go for Match Type 'Exact' in the left-hand column). Neither offer much volume though, so the results might not be that accurate.
In your title tag (and everywhere), I think you should use whatever makes most sense for the user, with SEO secondary. Your text should look and read naturally otherwise it'll come across as spammy.
It's good to use variations of your keywords, and not always right next to each other, e.g. "plumbers from Austin". Take a look at this for example: http://www.seomoz.org/blog/perfecting-keyword-targeting-on-page-optimization - it's an old-ish post, you might not want to include as many repetitions of the keywords - natural text is the key.