Bankruptcy Law Questions? Ask a Bankruptcy Lawyer.
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If you personally signed for the debt, you can be held accountable for it, even if the company is out of business. Unfortunately, the credit card company, as you have found out, has no obligation to accept a settlement. If you make it attractive enough they may, but they may choose to take judgment against you which they can enforce pursuant to the laws of your state. Some states allow liens against personal property, levies on personal bank account and/or wage garnishment to satisfy a judgment.
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