Example 1 Given real numbers $x, y$ satisfy
$$
3|x+1|+2|y-1| \leqslant 6 \text {. }
$$

Then the maximum value of $2 x-3 y$ is $\qquad$ (1)
Solve As shown in Figure 1, the figure determined by inequality (1) is the quadrilateral $\square A B C D$ and its interior, enclosed by four straight lines, where,
$$
\begin{array}{l}
A(-1,4), B(1,1), \\
C(-1,-2), D(-3,1) .
\end{array}
$$

Consider the family of lines $2 x-3 y=k$, i.e.,
$$
y=\frac{2}{3} x-\frac{k}{3} \text {, }
$$

which has the smallest intercept $-\frac{4}{3}$ at the lowest point $C$ of $\triangle A B C D$.
Therefore, when $x=-1, y=-2$, $2 x-3 y$ attains its maximum value of 4.