Take another look at the original sentence:

Bored with the lecture on comma splices and fused sentences, Jayson started poking Kim with the sharp tip of his pencil finally she whirled around and slugged him in the mouth.

You chose to fix it the correct way:

Bored with the lecture on comma splices and fused sentences, Jayson started poking Kim with the sharp tip of his pencil; finally, she whirled around and slugged him in the mouth.

Jayson started poking Kim with the sharp tip of his pencil is a main clause. She whirled around and slugged him in the mouth is a second main clause. A semicolon is an excellent way to join two main clauses that are closely related in meaning.

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