3. For $i=1,2, \cdots, n$, we have $\left|x_{i}\right|<1$, and
$$
\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=19+\left|x_{1}+x_{2}+\cdots+x_{n}\right| \text {. }
$$

Find the minimum value of the positive integer $n$.
Given $19=\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|-\mid x_{1}+x_{2}+\cdots+x_{n}|\leqslant| x_{1}|+| x_{2}|+\cdots+| x_{n} \mid<n$, it follows that $n \geqslant 20$. When $n=20$, take
$$
x_{i}=\left\{\begin{array}{cc}
0.95, & \text { when } i \text { is odd: } \\
-0.95, & \text { when } i \text { is even. }
\end{array}\right.
$$

They satisfy the given conditions, hence the minimum value of $n$ is 20.