I have two systems of inequalities with $3$ variables
The first system of inequality: $$ \left \{ \begin{align*} ax &> x + y + z \\ by &> x + y + z \\ cz &> x + y + z \end{align*} \right. $$
where $a,b,c,x,y,z>0$
The second system of inequality: $$ \left \{ \begin{align*} ax + by &> x + y + z \\ ax + cz &> x + y + z \\ by + cz &> x + y + z \end{align*} \right. $$ where $a,b,c,x,y,z > 0$
For both systems, I would like to find the relationship between $a$, $b$ and $c$ such that the system of inequalities has a solution.
For instance, the system of inequalities with $2$ variables below: $$ ax > x + y \\ bx > x + y \\ a,b,x,y > 0 $$ Can be easily reduced to $ab > a + b$,
that is the equation has a solution whenever $ab > a + b$
I would like to reduce the above system of equalities with $3$ variable $(x, y, z)$ in similar fashion.
I have searched online for system of inequalities with $3$ variables but am not coming up with anything much.