# Solving Inequalities using Mathematica

How can I solve this inequality using Mathematica?

Sqrt[B^2 + J^2] Cosh[(2 J)/T ] < J Sinh[  (2 Sqrt[B^2 + J^2])/T]


where T, J, and B are reals and they are >0.

• I doubt that something that complicated can be solved analytically at all. Reduce[ { Sqrt[B^2 + J^2] Cosh[(2 J)/T] < J Sinh[(2 Sqrt[B^2 + J^2])/T], {B, J, T} > 0}, {B, J, T} ]  quickly returns saying that "This system cannot be solved with the methods available to Reduce". – MarcoB Jul 7 '20 at 18:42

You can divide both sides by T. This reduces the number of variables to 2: $$j=\frac{J}{T},\quad b=\frac{B}{T}.$$
To visualize inequality, one can use ContourPlot. The inequality is fulfilled in the green area.
ContourPlot[Sqrt[b^2 + j^2] Cosh[2 j] - j Sinh[2 Sqrt[b^2 + j^2]],{j,0,1},{b,0,1},