How long before I give up on a keyword
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Making the assumption that all "controllable" factors are done correctly, and your DA and PA are competitive, is there a good rule of thumb on how long I should give a keyword before I try for a different keyword if I'm unable to break the first page on the Google's SERP?
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No, there is no rule of thumb. It's all about your goals, budget, patience, requirements, etc.
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Actually why don't you give a try for another keyword while still working on the old one? Never concentrate on just one keyword, there are a lot of fish in the sea

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I don't know what keyword you are targeting but I will say that there are many that I would not attempt because I don't have the resources to compete.
However, I very often have attacked a difficult term, never achieved it, but dominated nearly all of the longtail - enough that it was very worthwhile.
I like media/s suggestion to attack multiple keywords.
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Target a variation of the keyword by adding on a additional modifier. You should first target this longer tailed keyword before targeting the more competitive one.
For example, I created a site targeting "law firm seo"… since this was very competitive, I started targeting "law firm seo services" first and reached #1. And, now the site also ranks #1 for "law firm seo" as a result of the quality content, marketing and social engagement.