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Tuza's Conjecture states that if a graph $G$ does not contain more than $k$edge-disjoint triangles, then some set of at most $2k$ edges meets alltriangles of $G$. We prove Tuza's Conjecture for all graphs $G$ having nosubgraph with average degree at least $7$. As a key tool in the proof, weintroduce a notion of reducible sets for Tuza's Conjecture; these aresubstructures which cannot occur in a minimal counterexample to Tuza'sConjecture. We also introduce weak K\"onig--Egerv\'ary graphs, a generalizationof the well-studied K\"onig--Egerv\'ary graphs.

Tuza's Conjecture for Graphs of Maximum Average Degree Less Than 7