Example 1. Solve the absolute value equation
$$
|x+1|+|x+2|+|x-5|=10 .
$$
$$
\begin{array}{c}
(-2-x)+(-1-x)+(5-x)=10, \\
\therefore=-\frac{8}{3} .
\end{array}
$$
$$
\begin{array}{c}
(x+2)+(2+1)+(5-x)=10, \\
x=2 .
\end{array}
$$
$\therefore$ The solutions are: $x_{1}=-\frac{8}{3}, x_{2}=2$.
Analysis: Transform the problem into visual thinking, which is to find the point $x$ on the number line that is a total distance of 10 from the points $-1, -2,$ and $5$ (as shown in Figure 1).