Checking the existence of a solution for a set of linear equality and ineaulity equations

I would like to know if there is a method to check the existence of the solution for a given set of linear equations composed with both equalities and inequalities? I'm not interested in the solution, but whether there exist at least one solution or not.

Edit1: Mixed systems of linear equations and inequalities,

$a_{1,1} x_1 + \dots + a_{1,n} x_n = b_1$

$a_{2,1} x_1 + \dots + a_{2,n} x_n = b_2$

$\dots$

$a_{m-1,1} x_1 + \dots + a_{m-1,n} x_n \geq b_{m-1}$

$a_{m,1} x_1 + \dots + a_{m,n} x_n \geq b_m$

The number of equations $m$ might be less than, equal to or greater than $n$ (the number of unknowns).

Edit2: Example1,

$\begin{cases} 2x-y \geq -3 \\ -4x-y \geq -5 \\ -x+y=4 \end{cases}$

There is no solution for above system. I'm looking for a systematic (algorithmic) way to determine if such systems of equations have any solution or not.