A way for federal courts to hear claims for which they would not ordinarily have jurisdiction. Supplemental jurisdiction only exists in the situation where a lawsuit consists of more than one claim, and the federal court has valid jurisdiction (either diversity jurisdiction or federal question jurisdiction) over at least one of the claims. In that situation, if the federal claim and the other claims arise out of a "common nucleus of operative fact," then the court may (but does not have to) exercise supplemental jurisdiction to hear the other claims as well. United Mine Workers v Gibbs, 383 US 715 (1966).
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