Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we consider are a conjunction of a linear system Ax = b and a conjunction of non-convex constraints of the form $x_i \le x_j^n$. We show that the satisfiability of these constraints is in NP. As a consequence, we obtain NP completeness for an extension of quantifier-free constraints on sets with cardinalities (QFBAPA) with function images $S = f[P^n]$. We also present related hardness results and consequences of complexity for dual, convex, constraints.