Determining for what values is a system of inequalities true.

Question

I have the following system of inequalities:

\begin{equation} 8A^2\alpha+44A\alpha+60\alpha>3A^2\alpha^2+12A^2+18A\alpha^2+40A+27\alpha^2+40 \tag{1} \end{equation} \begin{equation} A > 0 \tag{2} \end{equation} \begin{equation} \alpha > 1 \tag{3} \end{equation}

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

UPDATE: Instead of determining whether it is true/false, wanted to figure out for what range(s) of values of $\alpha$ is the system of inequalities true. If $(1$) and $(2)$ is true for $\alpha \le 1$ then that will mean no solution exists.

UPDATE 2: This is just an algebra problem at this point. Testing at $\alpha = 1$ is false and finding when RHS is equal to LHS has no solution for $\alpha > 1$. Therefore there does not exist an $\alpha > 1$.

Answer 1

If $A=1$ and $\alpha=2$, then the left side is $224$ and the right side is $284$. So the inequality is false.

Answer 2

LHS is $\alpha (8A^2 + 44A+60)$ and RHS is $\alpha^2(3A^2 +18A + 27) +(12A^2 +40A + 40)$. And as all components are positive:

So $\alpha (8A^2 + 44A+60) > \alpha^2(3A^2 +18A + 27) +(12A^2 +40A + 40) \iff$

$\frac {8A^2 + 44A+60}{3A^2+18A+27} - \frac 1{\alpha}\frac {12A^2 +40A + 40}{3A^2+18A+27} > \alpha$.

So this will be false whenever $\alpha \ge \frac {8A^2 + 44A+60}{3A^2+18A+27} - \frac 1{\alpha}\frac {12A^2 +40A + 40}{3A^2+18A+27}$ which will include (but not be restricted to) whenever $\alpha \ge \frac {8A^2 + 44A+60}{3A^2+18A+27}$.

So....

For any $A > 0$ we can always find $\alpha \ge \frac {8A^2 + 44A+60}{3A^2+18A+27}$ is which case $\alpha (8A^2 + 44A+60) < \alpha^2(3A^2 +18A + 27) +(12A^2 +40A + 40)$

====

Alternatively bases on achille hui's comment:

$8A^2\alpha+44A\alpha+60\alpha>3A^2\alpha^2+12A^2+18A\alpha^2+40A+27\alpha^2+40 \iff$

$0>-3A^2a^2 > (12A^2 + 18A\alpha^2 +40A + 27\alpha^2 +40) -(44A\alpha +60\alpha)$

It's pretty clear that if we take $A$ and $\alpha$ large enough we can find values where the RHS is positive.

For example: If we let $18A\alpha^2 > 44A\alpha$ or in other words let $\alpha > \frac {44}{18}=\frac {22}9$ we have:

$ (12A^2 + 18A\alpha^2 +40A + 27\alpha^2 +40) -(44A\alpha +60\alpha)>$

$(12A^2 + 44A\alpha +40A + 27\cdot \frac {22}{9}\alpha +40)-(44A\alpha +60\alpha)=$

$12A^2 +40A + 66\alpha +40)-60\alpha=$

$12A^2 + 40A + 6\alpha + 40$

Which clearly positive.