  For several embedded surfaces with zero self-intersection number in
4-manifolds, we show that an adjunction-type genus bound holds for at least one
of the surfaces under certain conditions. For example, we derive certain
adjunction inequalities for surfaces embedded in $m\mathbb{CP}^2\#
n(-\mathbb{CP}^2)$ ($m, n \geq 2$). The proofs of these results are given by
studying a family of Seiberg-Witten equations.
