The paper deals with the algebraic Riccati equation \[ (1)\quad -{\mathcal R}(X):=XBR^{-1}B^*X-X(A-BR^{-1}C)-(A-BR^{-1}C)^*X-(Q- C^*R^{- 1}C)=0, \] where (A,B) is stabilizable. The main results concern the interplay between equation (1) and the corresponding Riccati inequality \({\mathcal R}(X)\geq 0\). Define \(M:=\{X| X=X^*,\quad {\mathcal R}(X)\geq 0\}\) and \(N:=\{X| X=X^*,\quad {\mathcal R}(X)=0\}.\) Theorem 2.1: Assume \(M\neq \emptyset\). Then there exists an \(X_+\in N\) such that \(X_+\geq X\) for all \(X\in M\). Then in particular \(X_+\) is the maximal Hermitian solution of (1). Moreover, all the eigenvalues of the matrix \(A-BR^{-1}(C+B^*X_+)\) are in the closed left half plane. The proof provides an iterative procedure for obtaining the maximal Hermitian solution of (1) (as usual, \(X\geq Y\) for Hermitian matrices means that X-Y is a positive semidefinite matrix). As a consequence of this result a comparison of the maximal solutions of two Riccati equations is given. Both the continuous-time case and the discrete-time case are considered. The discrete-time case is in many ways the same as the continuous one.