How to find out if any point of a 2d line segment satisfies a system of linear inequations of degree 1 with one parameter? Non formally, I want to know how to find out if a line segment intersects the area bounded by three axis-parallel lines. Line segment is not necessarily axis-parallel. I could always bruteforce a point if there weren't an infinity of them.
Edit: here's an example image for clarification. Black segments intersect the area, green segments don't.

 A: If I understood your problem, we have a matrix $A\in \Bbb R^{n\times 2}$ and $v,w\in\Bbb R^{2},c \in \Bbb R^n$ and we need to know if the line segement $$s:[0,1]\to \Bbb R^2:t\mapsto v+wt$$ satisfies $As(t_0)\geq c$ for one $t_0\in [0,1]$. Here the inequality is taken component-wise and $A,c$ are chosen to describe the constraints (you want to know if $s(t)$ is in $\{x\mid Ax \geq c\}$). 
We treat three cases:


*

*If $As(0)\geq c$ then $t_0=0$

*If $As(1)\geq c$ then $t_0=1$

*If $As(0)\ngeq c$ and $As(1)\ngeq c$, note that $\require{cancel} As(t)\ngeq c \cancel{\iff} As(t)<c$ since we are working with component-wise inequalities. There is $t_0\in ]0,1[$ such that  $As(t_0)\geq c$ if and only if the constraints that were violated in $As(0)\ngeq c$ are now satisfied in $As(1) \ngeq c$. Formally, let $I_0 = \{i\mid (As(0))_i<c_i\}$ and $I_1 = \{i\mid (As(1))_i\geq c_i\}$. If $I_0 \subset I_1$, then such a $t_0$ exists.
Edit:(construction of the matrix $A$ and $c$)
If you have the system of inequalities
$$ a_{1,1}x+a_{1,2}y \geq c_1 \\ a_{2,1}x+a_{2,2}y \geq c_2 \\ \vdots \\ a_{n,1}x+a_{n,2}y \geq c_n $$
It can be rewritten $$A\begin{pmatrix} x\\ y \end{pmatrix}\geq \begin{pmatrix}c_1\\c_2\\ \vdots \\ c_n \end{pmatrix}$$
With $A\in\Bbb R^{n\times 2}$ defined by $A_{i,j}=a_{i,j}$. In your example
$$A = \begin{pmatrix}1 & 0\\ -1 & 0 \\ 0 & 1\end{pmatrix} \text{ and } c=\begin{pmatrix}5 \\ -8 \\ 7 \end{pmatrix}$$
