I have two linear functions:
f(x) = 2x - m + 6
g(x) = -x + 2m + 3
I have to prove that for all m belonging to real numbers they intersects in C(x,y), and the coordinates must satisfy below inequation:
|x - 1| - |5 - y| <= 1
I did equation system and solved it using determinants, and my result is:
x = 4m - 16
y = 4m - 20
So point C will be C(4m - 16, 4m - 20).
Determinants have one solution when main determinant is not equal to 0. I solved it and got 3 != 0, so for all real m there is point of intersect. So I have proved first thing.
But I'm stuck on proving that coordinates of C satisfies inequality I mentioned earlier. I know it must be true, but when I try to prove it then some inequations occurs to be false (and I think they all will be true, maybe I'm wrong?). I tried to do it by writing absolute values as piecewise.
The coordinates of the point of intersection are $(m - 1, m + 4)$ (instead of the values that you quote in your post).
With these values, you can find the following results for the inequation : \begin{cases} 1 & \text{for } m \le 1\\ 3 - 2m & \text{for } 1 \lt m \lt 2\\ -1 & \text{for } m \ge 2 \end{cases}
from which you can easily prove that the inequality is indeed satisfied.