So I'm trying to figure out if it's possible to extract the private key if you have the ciphertext and the IV corresponding to that ciphertext. It's for some kind of a challenge, and can't really seem to figure it out.
No, that's not really possible without blatant flaws of the implementation. Modern modes of operation of ciphers are resistent to attacks even if you know many pairs of plaintext and ciphertext - and the IV is public knowledge. Knowing it is the normal case. You also didn't mention what operation mode was used.
Well, of course you could brute force the key, but that would take more time and computational power than you can ever muster.
Oh, and a small note: AES is a symmetric cipher, so there's just one key, which is normally not called private key. It may be true, but this term is generally reserved for asymmetric encryption.