\begin{align*} ax > b & \iff ax \geq b + \epsilon \\ ax < b & \iff ax \leq b - \epsilon \\ ax = b & \iff ax \leq b \land ax \geq b \\ ax \geq b & \iff -ax \leq -b \\ ax \leq b & \iff -ax \geq -b \\ \end{align*}