If you could edit your post to include your inequalities, or at least similar inequalities that would demonstrate an equivalent problem, and describe what form you need the solution to be in then readers might be able to experiment with this and see if they can find you a solution.

Since you describe the boundaries of inequalities, would finding a solution to equalities or subsets of those equalities be close to what you are looking for?

Please check this very carefully to see if I have made any mistakes

x1=15.75*10^-6; a=x1 x2 x3^3; b=x1 x3^3; c=(9 x1+x3)x3^2; d=36 x1 x3+3(3+x4/x5)x3^2;
e=60 x1+12(3-2 x4/x5)x3; f=60(1+x4/x5); X=b c-a d; Y=b e-a f; Z=(d X-b Y)Y-X^2 f;
Con1=d X Y-(b Y^2+f X^2); Con2=(5 x1+3 x3)/(2 x3);
NMaximize[{Con1, Con2>x4/x5 && 1*10^-6<x2<4*10^-6 && 20*10^-6<x3<50*10^-6 &&
  10<x4<100 && 10<x5<100},{x2,x3,x4,x5}]
(*{7.367860243792591*^-59, {x2 -> 3.0786924361325835*^-6,
   x3 -> 0.00004972976008268178, x4 -> 10.031234356582235,
   x5 -> 44.64769318375559}}*)

Con2>x4/x5 /. %[[2]]
(*True*)

Because that value of Con1 is so small it may be nothing more than roundoff error. It may also indicate how difficult it may be to find the boundary where Con1 > 0.