Question 22
Let $S = x + y + z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities:
$$
\left\{\begin{array}{rl}
x y z & \gt 1 \\
x + y + z & \gt \frac{1}{x} + \frac{1}{y} + \frac{1}{z}
\end{array} .\right.
$$
Prove that $S$ can take on any real values when $x, y, z$ vary.
$$
\left\{\begin{array}{rl}
x y z & \gt 1 \\
x + y + z & \gt \frac{1}{x} + \frac{1}{y} + \frac{1}{z}
\end{array} .\right.
$$
Prove that $S$ can take on any real values when $x, y, z$ vary.