# Can a linear programming method be used to solve systems of inequalities with OR (disparate) compound inequalities?

I recently discovered linear programming and it seemed perfect for a CS problem I wanted to solve a few months ago. This task involved solving a large quantity of inequalities at once.

For example, one such system could be

X1,X2,X3,X4 >= 0
X1 >= X3 + 3.0 or X1 < X3 - 7.2
X2 >= X4 + 5
X2 >= X3 + 3.0 or X2 < X3 - 5.3
X2 >= X1 + 7.2 or X2 < X1 - 5.3


And then some reasonably complex objective function in terms of X1, X2, X3, X4. In fact, the function mostly just needs to aim to reduce the spread of the variables, whilst finding a solution.

(Optimally, the program would aim to find a solution that meets as many inequalities as possible, to allow it to find a "close enough" solution when there is no actual solution)

I can't find anyone else talking about the limitations of linear programming, but I understand these inequalities are only "kinda" linear. If anyone out there, perhaps with a formal higher CS education, I'd much appreciate it.

If this is not possible with normal linear programming algorithms, any leads into possible solutions would be much appreciated.