  The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and
the colored Jones polynomial of $K$. If a two-bridge knot $K$ satisfies the
$AJ$-conjecture, we give sufficient conditions on $K$ for the $(r,2)$-cable
knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram
of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then
$C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the
conditions of the main theorem are satisfied.
