Example 1 (1994 National High School League Question) Given the directed line segment $P Q$ with the coordinates of the starting point $P$ and the end point $Q$ being $(-1,1)$ and $(2,2)$ respectively, if the line $l: x+m y+m=0$ intersects the extension of $P Q$, then the range of values for $m$ is $\qquad$ .
Solve: Fill in $-3<m<-\frac{2}{3}$. Reason: Clearly, the line passes through point $M(0,-1)$. The slope of the line $l_{1}$ passing through $M$ and parallel to $P Q$ is $K_{l_{1}}=K_{\mathrm{PQ}}=\frac{2-1}{2-(-1)}=\frac{1}{3}$, and the slope of the line $l_{2}$ passing through $M$ and $Q$ is $K_{l_{2}}=\frac{2-(-1)}{2-0}=\frac{3}{2}$. Therefore, the necessary and sufficient condition for $l$ to intersect the extension of $P Q$ is
$$
K_{l_{1}}<K_{l}<K_{l_{2}} \Leftrightarrow \frac{1}{3}<-\frac{1}{m}<\frac{3}{2} \Leftrightarrow-3<m<-\frac{2}{3} .
$$