jmc

We look at the $x$-intercept and the $y$-intercept. Since $y=-2(x-9)$, at $x=0,y=18$, and at $x=9,y=0$, the $y$-intercept is $(0,18)$ and the $x$-intercept is $(9,0)$. In order to keep both $x$ and $y$ integral, we investigate further. Since the slope is $-2$, a negative integer, and $y$ must be a nonnegative integer, $x-9$ must be a nonpositive integer. This means that all the integer $x$ values between 0 and $9$, inclusive are valid, since $x\le 9$ and so $x-9\le 0$. Therefore, there are $\boxed{10}$ total lattice points.