How to solve a system of linear inequalities? I am working on the following exercise:

Find a solution to the following system or prove that none exists:
\begin{align}
x_1-x_2 &\le 4\\
x_1-x_5 &\le 2\\
x_2-x_4 &\le -6 \\
x_3-x_2 &\le 1 \\
x_4-x_1 &\le 3 \\
x_4-x_3 &\le 5\\
x_4-x_5 &\le 10 \\
x_4-x_3 &\le -4 \\
x_5-x_4 &\le -8
\end{align}

I do not know how to do that. Is there some algorithm? I can not find anything online. Could you help me?

From the first inequality you have $$x_1 \leq 4+x_2.$$
Now from the second $$4+x_2-x_5 \leq 2$$ or $$ x_2-x_5 \leq -2$$ or $$x_2 \leq -2+x_5.$$
Now from the third you get $$ -2+x_5-x_4\leq -6$$ or $$x_5\leq -4+x_4.$$ This can be applied to the last inequality and you get $$-4+x_4-x_4\leq -8$$ and that gives $$-4 \leq -8.$$ So the system has no solution.
