How do I record the purchase of an existing business in QBO if I am a single member LLC and only purchased the assets of the existing business?

"inventory of $10,000"

Post inventory by actual count and name, if you intend to manage this as Inventory Items. Otherwise, that is an asset debit, if you are doing generic or periodic inventory management.

"equipment of $10,000, furniture of $15,000, and goodwill of $15,000"

Those are all Debits to various asset accounts. Equipment and Furniture is Fixed Assets. Goodwill is Other Asset.

The total Credit is Equity or the Bank payment (a check).

Ask your own CPA to help guide you.