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I don't think you're going to find any convenient public transportation that would take you from Ellicott City or Cockeysville to the western 'burbs. From Cockeysville, you might be able to take the light rail to downtown and then transfer to the 15, 40, or 20 bus line to Security Square Mall, but that would be a lengthy trip and you might still have to walk depending on where your office is located.
If you don't drive or don't like to drive, I'd suggested finding something closer to where you work.
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