Nephew 8. Find the number of lattice points inside and on the boundary of the triangle formed by the line $y=\frac{2}{3} x-\frac{1}{2}, x=10$ and the $x$-axis.
Given the function $y=\frac{2}{3} x-\frac{1}{2}$, when $x$ takes the values $1,2, \cdots$, 10 (note: when $x=\frac{3}{4}$, $y=0$), the corresponding $y$ values are $\frac{1}{6}, \frac{5}{6}, \frac{3}{2}, \frac{13}{6}, \frac{17}{6}, \frac{7}{2}, \frac{25}{6}, \frac{29}{6}, \frac{11}{2}, \frac{37}{6}$. The number of integer points on the known triangle (including the boundary) is equal to the sum of the integer parts of the $y$-coordinates plus the 10 integer points on the $x$-axis, i.e.,
$$
\begin{array}{c}
{\left[\frac{1}{6}\right]+\left[\frac{5}{6}\right]+\left[\frac{3}{2}\right]+\left[\frac{13}{6}\right]+\left[\frac{17}{6}\right]+} \\
{\left[\frac{7}{2}\right]+\left[\frac{25}{6}\right]+\left[\frac{29}{6}\right]+\left[\frac{11}{2}\right]+\left[\frac{37}{6}\right]+10=37}
\end{array}
$$

Therefore, there are 37 integer points inside and on the boundary of the triangle.