Linear system of inequalities/Remark

A system of linear inequalities over the rational numbers or over the real numbers is a system of the form

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&\star &c_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&\star &c_{2}\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&\star &c_{m}\,,\end{matrix}}$

where ${}\star$ might be ${}\leq$ or ${}\geq$. It is considerably more difficult to find the solution set of such a system than in the case of equations. In general, it is not possible to eliminate the variables.