Determining for what values is a system of inequalities true.

$\begin{array}{}\text{(1)}& 8{A}^{2}\alpha +44A\alpha +60\alpha >3{A}^{2}{\alpha}^{2}+12{A}^{2}+18A{\alpha}^{2}+40A+27{\alpha}^{2}+40\end{array}$

$\begin{array}{}\text{(2)}& A>0\end{array}$

$\begin{array}{}\text{(3)}& \alpha >1\end{array}$

Is there a procedure that determines for what value range(s) of $\alpha $ inequalities (1) and (2) are satisifed?

$\begin{array}{}\text{(1)}& 8{A}^{2}\alpha +44A\alpha +60\alpha >3{A}^{2}{\alpha}^{2}+12{A}^{2}+18A{\alpha}^{2}+40A+27{\alpha}^{2}+40\end{array}$

$\begin{array}{}\text{(2)}& A>0\end{array}$

$\begin{array}{}\text{(3)}& \alpha >1\end{array}$

Is there a procedure that determines for what value range(s) of $\alpha $ inequalities (1) and (2) are satisifed?