jmc

The left-hand side satisfies $$|x_1|+|x_2|+\cdots +|x_n|<1+1+\cdots +1=n,$$while the right-hand side satisfies $$19+|x_1+x_2+\cdots +x_n|\ge 19.$$Therefore, $n>19,$ so $n\ge 20.$ It is possible that $n=20,$ since, for example, we can choose $$\begin{array}{rl}{x}_1=x_2=\cdots =x_{10}& =\frac{19}{20},\\ x_{11}=x_{12}=\cdots =x_{20}& =-\frac{19}{20},\end{array}$$which makes $|x_1|+|x_2|+\cdots =|x_{20}|=19$ and $|x_1+x_2+\cdots +x_{20}|=0.$ Therefore the answer is $\boxed{20}.$