x*exp(x) and x*exp(-x) are both convex for x >= 0. You haven’t told us which of X and X* are CVX variables. In order for the constraint to be convex, the RHS has to be concave, which looks rather dubious.
As for chaining inequalities together, it is true that the CVX 2.2 Users’ Guide says
The latest version of CVX now allows you to chain inequalities together; e.g., l <= x <= u. (Previous versions did not allow chained inequalities.)
The CVX 1.22 Users’ Guide from August 2012 says
The latest versions of cvx now allow you to chain inequalities together; e.g., l <= x <= u.
So the CVX 2.2 Users’ Guide statement
Previous versions did not allow chained inequalities
is not true.
The issue for this question is not the chaining, but the convexity of the constraints.