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My current job is going out of business in a few months. I received and signed a job offer contingent on a satisfactory background check. I have no criminal background, but did not want to give notice to my current employer until the background check comes back. I reached out to the HR person about what paperwork is needed to get the ball rolling on the background check. I received an email reply back from HR stating for my position a background check is not required. Is it alright for me to give notice now?