We consider classical linear and quasilinear elliptic inequalities as well as divergence structure and variational operators, with emphasis on the important topics of comparison results and tangency theorems. This work ultimately applies also to weak solutions in appropriate Sobolev spaces. In order that the book may serve the purposes of reference and as a basis for further developments, the proofs are given in detail. This had led, at a number of points, to results either not found elsewhere, or not readily accessible. Many of the proofs and derivations, even of the standard parts of the theory, are new, along with the first book presentation of the modern compact support principle and the general theory of structured elliptic inequalities. The proofs here, though difficult, make the subject available for the first time to the the general reader.