Re-writing the first equation:

2(8 + x) = 34 + x
16 + 2x = 34 + x
2x - x = 34 - 16
x = 18

Therefore, x = 18

Re-writing the second equation:

120 : x - 120 : (x + 6) = 1
120 : 18 - 120 : (18 + 6) = 1
120 : 18 - 120 : 24 = 1
6 : 1 = 1

Therefore, the solution to the second equation is 6 : 1 = 1.