Trying to optimize ratios of variables is a common problem that crops up in practice. While this fractional constraint can not be directly addressed with linear programming, it can be reformulated with the Charnes-Cooper transformation, given the following fractional linear programming problem. (Side note, this is based on a point where $d^Tx = \beta$ is not feasible. Otherwise, it would be unbounded).
\[\max_x \frac{c^Tx+\alpha}{d^Tx + \beta}\] \[\text{s.t. } Ax\leq b\]We can use the variable transform to create a linear program without the fractional objective.
\[t = \frac{1}{d^Tx + \beta}\] \[y = tx = \frac{1}{d^Tx + \beta}x\]The resulting programming problem is just a particular linear programming problem.
\[\max_{y, t} c^Ty + \alpha t\] \[\begin{align*} \begin{split} \text{s.t. }d^Ty + \beta t &= 1\\ Ay - bt &\leq 0\\ -t&\leq 0 \end{split} \end{align*}\]The solution to the fractional optimization problem is $x = \frac{y}{t}$. I ran into this program when looking at planning problems in my studies. This transformation lets you solve fractional linear programming problems with any linear programming solver.