  Tuza conjectured that for every graph $G$, the maximum size $\nu$ of a set of
edge-disjoint triangles and minimum size $\tau$ of a set of edges meeting all
triangles satisfy $\tau \leq 2\nu$. We consider an edge-weighted version of
this conjecture, which amounts to packing and covering triangles in
multigraphs. Several known results about the original problem are shown to be
true in this context, and some are improved. In particular, we answer a
question of Krivelevich who proved that $\tau \leq 2\nu^*$ (where $\nu^*$ is
the fractional version of $\nu$), and asked if this is tight. We prove that
$\tau \leq 2\nu^*-\frac{1}{\sqrt{6}}\sqrt{\nu^*}$ and show that this bound is
essentially best possible.
