Property 3 forces that the solution be, approximately, x=1695 and z=0.28.
If instead you have properties 1 and 2, you can deduce a bit more as follows.
1800<x<2170 implies z+1=2170/Interval[{1800, 2170}] which is Interval[{1, 217/180}]. So 0<z<37/180.
