  We consider parametrized systems of generalized polynomial inequalities (with
real exponents) in $n$ positive variables $x$ and involving $m$ monomial terms
$c \circ x^B$ (with exponent matrix $B$, positive parameters $c$, and
component-wise product $\circ$). More generally, we study solutions $x \in
\mathbb{R}^n_>$ to $(c \circ x^B) \in C$ for a "coefficient cone" $C \subseteq
\mathbb{R}^m_>$. Our framework encompasses systems of generalized polynomial
equations, as studied in real fewnomial and reaction network theory.
  We identify the relevant geometric objects of the problem, namely a bounded
set $P$ arising from the coefficient cone and two subspaces representing
monomial differences and dependencies. The dimension of the latter subspace
(the monomial dependency $d$) is crucial.
  As our main result, we rewrite the problem in terms of $d$ binomial
conditions on the "coefficient set" $P$, involving $d$ monomials in the
parameters. In particular, we obtain a classification of polynomial inequality
systems. If $d=0$, solutions exist (for all $c$) and can be parametrized
explicitly, thereby generalizing monomial parametrizations (of the solutions).
Even if $d>0$, solutions on the coefficient set can often be determined more
easily than solutions of the original system.
  Our framework allows a unified treatment of multivariate polynomial
inequalities, based on linear algebra and convex geometry. We demonstrate its
novelty and relevance in three examples from real fewnomial and reaction
network theory. For two mass-action systems, we parametrize the set of
equilibria and the region for multistationarity, respectively, and for
univariate trinomials, we provide a "solution formula" involving discriminants
and "roots".
