I want to solve two inequations to find a variable range but Mathematica starts running but doesn't show any results after a long time.

First one should be NPV,s,Sale < NPV,s,Service. It will show us the r_min: enter image description here

First one should be NPV,c,Service < NPV,c,Sale. It will show us the r_max:

enter image description here

there is no other equation or inequation.

I used this code in Mathematica but I didn't get any results.

enter image description here

Solve[(1 - Ss)*(P - A) < (1 - Ss)*Sum[r/qs^t, {t, 1, T}] - A + 
     Ss*Sum[(bs*A)/qs^t, {t, 1, T}] + (1 - Ss)*Rt*(qs^(-T)) && 
   -P + Sc*Sum[(bc*P)/qc^t, {t, 1, T}] + ((1 - Sc)*Rt)/qc^T > 
    (-(1 - Sc))*Sum[r/qc^t, {t, 1, T}], r]

Factor the r from the Sum's and replace each sum with a constant:

res1=Reduce[(1 - Ss)*(P - A) < (1 - Ss)*r s1 - A + 
    Ss*s2 + (1 - Ss)*Rt*(qs^(-T)) && -P + 
    Sc*s3 + ((1 - Sc)*Rt)/qc^T > (-(1 - Sc))*r s4, r]

this produces an enormous conditional expression. You can try to simplify by adding appropriate assumptions on the symbols, eg,

Simplify[res1,Assumptions->{Element[{A, P, qc, qs, r, Rt, s1, s2, s3, s4, Sc, Ss, T},Reals],A>0}]

this is taking a long time, but if you know more appropriate assumptions to add on your constants it may help.

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