# System of inequalities with assumptions

I have the following problem:

I have a Variable a for which I know that -5<a<-2 is valid.

Furthermore, I have the following two equations:

EV1 = -0.5*b + 0.5*Sqrt[b^2 - 4*c]
EV2 = -0.5*b - 0.5*Sqrt[b^2 - 4*c]


with

b = -a + 3.4 + L2
c = (a + 0.2)*(3.2 + L2) - 0.25*(-9.5 - L1)^2


Based on those equations, I want to compute an L1>2 and an L2 such that EV1<-2 and EV2<-2 for all a in the defined interval -5<a<-2.

\$Assumptions = -5 < a < -2
(*1*)
Simplify[Reduce[EV1 < -2 && EV2 < 2 && L1 >= 2, {L1, L2}]]
(*2*)
Simplify[FindInstance[EV1 < 1.7 && EV2 < 1.7 && L1 >= 2, {L1, L2}]]


Unfortunately, both didn't work.

Does anybody know a possible way to compute an instance for L1 and L2 fulfilling the requirements?

If you use rational numbers in your variables and equations, Reduce works and shows you, that you conditions can't get fullfilled.

      b = Rationalize[-a + 3.4 + L2, 0]
c = Rationalize[(a + 0.2)*(3.2 + L2) - 0.25*(-9.5 - L1)^2, 0]

EV1[L1_, L2_] = -1/2*b + 1/2*Sqrt[b^2 - 4*c] // Simplify
EV2[L1_, L2_] = -1/2*b - 1/2*Sqrt[b^2 - 4*c] // Simplify

(red =
Reduce[EV1[L1, L2] < -2 && EV2[L1, L2] < -2 &&
L1 >= 2 && -5 < a < -2, {L1, L2}])

(*  False  *)