Solving each inequality:

1) 2+x > 8-x
3x > 6
x > 2

2) 2x+1 > x+6
x > 5

3) 6x+1 < 2x+9
4x < 8
x < 2

4) 7x+4(x-2) > 6(1+3x)
7x + 4x - 8 > 6 + 18x
11x - 8 > 6 + 18x
-7x > 14
x < -2

So the solutions are:
x > 2
x > 5
x < 2
x < -2