Find the condition under which an inequality holds

Question

I've got an inequality of 8 non-negative parameters (arising from a 5x5 matrix) and would like to find a condition under which my inequality holds. The condition would be an inequality too of these parameters. I'm not looking for the numerical values of the parameters, but trying to get an if and only if relation in a proof with a very large inequality to solve. For instance, given that all parameters are non-negative and 0<q<1, what condition should the parameters satisfy in order to have the following inequality:

      n*k*(β1*(1 - q) + β2*q)) - ((k + μ)*(γ1 + μ)*(γ2 + μ) - 
     β2*n*k*q*(γ1 + μ) - β1*n*k*(1 - q)*(γ2 + μ)) > 0)

Does anyone know if Mathematica can do this? Thank you.

Answer 1

First, don't use N as symbol, it's predfined in Mathematica

Try Reduce to solve the problem

expr = (n*
 k*(\[Beta]1*(1 - q) + \[Beta]2*
    q)) - ((k + \[Mu])*(\[Gamma]1 + \[Mu])*(\[Gamma]2 + \[Mu]) -\[Beta]2*n*k*q*(\[Gamma]1 + \[Mu]) - \[Beta]1*n*k*(1 - q)*(\[Gamma]2 + \[Mu])) ;
var = Variables[expr];
Reduce[Join[{expr > 0}, Map[# > 0 &, var]]] // Simplify


(*k > -((\[Mu] (\[Gamma]1 + \[Mu]) (\[Gamma]2 +\[Mu]))/(-n q \[Beta]2 \(1 + \[Gamma]1 + \[Mu]) + (\[Gamma]1 + \[Mu])(\[Gamma]2 + \[Mu]) + n (-1 + q) \[Beta]1 (1 + \[Gamma]2 +\[Mu]))) 
&&n > ((\[Gamma]1 + \[Mu]) (\[Gamma]2 + \[Mu]))/(q \[Beta]2 (1 + \[Gamma]1 + \[Mu]) - (-1 +q) \[Beta]1 (1 + \[Gamma]2 + \[Mu])) 
&& \[Beta]2 >0 && \[Gamma]1 > 0 && \[Gamma]2 > 0 && \[Mu]> 0 
&& ((\[Beta]1 > (\[Beta]2 (1 + \[Gamma]1 + \[Mu]))/(1 + \[Gamma]2 + \[Mu]) 
&&0 < q < (\[Beta]1 (1 + \[Gamma]2 + \[Mu]))/(-\[Beta]2(1 + \[Gamma]1 + \[Mu]) + \[Beta]1 (1 + \[Gamma]2 + \[Mu]))) || (q > 0 && 0 < \[Beta]1 <= (\[Beta]2 (1 + \[Gamma]1 + \[Mu]))/(1 + \[Gamma]2 + \[Mu])))*)