What happens when Inventory goes up by $10, assuming you pay for it with cash?

No changes to the Income Statement.

On the Cash Flow Statement, Inventory is an asset so that decreases your Cash Flow from Operations – it goes down by $10, as does the Net Change in Cash at the bottom.

On the Balance Sheet under Assets, Inventory is up by $10 but Cash is down by $10, so the changes cancel out and Assets still equals Liabilities & Shareholders’ Equity.