A problem related to intersecting lines if I save some line equations (y=ax+b) in the following way for each line, with two points:
(0,b),(1,a+b), and none og them intersect with another, for example:
$y=x+1 -> (0,1),(1,2)$
$y=x+2 ->(0,2),(1,3)$
$y=x+3 ->(0,3),(1,4)$
$y=2x-1 -> (0,-1),(1,1)$
and want to add another line, only if it doesn't intersect with my previous lines between $0<=x<= 1$
for example I want to add y=6x-1 ((0,-1),(1,5))
why is it enough to check if it intersects with the predecessor/successor in y value?
for example, here it is enough to check it with  y=x+1  
 A: This is a brutal question, but I enjoyed doing it quite a lot.
Let’s say the new line to be added is $$y=mx + c $$
which has two parameters. We’ll find out the set of solutions of $m,c$ that satisfy the given conditions.
Firstly,  get the $x$-coordinate  the point of intersection of this new line with the previous lines:
$$L_1 :\ x=\frac{1-c}{m-1} \\ L_2: x=\frac{2-c}{m-1} \\ L_3: x= \frac{3-c}{m-1} \\ L_4: x=\frac{1+c}{2-m}$$
where $L_1, L_2, L_3, L_4$ are lines respectively in the order specified in the question.
Now, we need each of these intersection abscissas to lie outside of the interval $[0,1]$, which is equivalent to saying
$$ x \gt 1\, \vee \, x\lt 0$$ in each case. Applying this condition yields $4$ different conditional inequalities in $m$ and $c$ which we must take the intersection of. Simplifying these we get the following:
$$\bullet \space m+c \lt 2 \, \vee \, (c \gt 1 \wedge m \gt 1)$$
$$\bullet \space m+c \lt 3 \, \vee \, (c \gt 2 \wedge m \gt 1)$$
$$\bullet \space m+c \lt 4 \, \vee \, (c \gt 3 \wedge m \gt 1)$$
$$\bullet \space m+c \gt 1 \, \vee \, (c \lt -1 \wedge m \lt 2)$$
It should be fairly easy to solve these inequations graphically. Here’s a very rough picture for the solution set. Note the $x$ and $y$ axes represent $c$ and $m$ respectively.

The yellow region is the solution set.
We can infer from the graph that the solution set is as follows for $m,c \in \mathbb R$:
$$ ( m \lt 2 \wedge c \lt -1)\, \vee \,
   ( 2 \lt m+c \lt 3 ) \, \vee \, (m \gt 1 \wedge c \gt 1 \wedge m+c \lt 3) \, \vee \, (m \gt 1 \wedge c \gt 2 \wedge m+c \lt 4) \, \vee \, (m \gt 1 \wedge c \gt 3)$$
