Example 1. Find the minimum value of $|x-1|+|x-3|+|x-5|$.
To solve this type of problem, we can use the method of "fixing points, dividing segments, and discussing."

Points. Let $x-1=0, x-3=0, x-5=0$, to determine $x_{1}=1, x_{2}=3, x_{3}=5$ three points.

Dividing segments. The above three points divide the number line into four segments: $(-\infty, 1]$, $(1,3]$, $(3,5]$, $(5,+\infty)$.

Discussion. When $x \in(-\infty, 1]$, the original expression $=9-3 x$. In this case, the minimum value of the original expression is 6;

When $x \in(1,3]$, the original expression $=7-x$. In this case, the minimum value of the original expression is 4;

When $x \in(3,5)$, the original expression $=x+1$. In this case, the minimum value of the original expression is 4;

When $x \in(5,+\infty)$, the original expression $=3 x-9$. In this case, the minimum value of the original expression is 6.

In summary, the minimum value of the original expression is 4.