Solving the first inequality:

2(x+6) - 5x > 0
2x + 12 - 5x > 0
-3x + 12 > 0
-3x > -12
x < 4

So, the solution for the first inequality is x < 4.

Solving the second inequality:

x/6 - x/2 < 1
To simplify, we can find a common denominator of 6 for both terms:

x/6 - 3x/6 < 1
-x/6 < 1
-x < 6
x > -6

So, the solution for the second inequality is x > -6.

Putting both solutions together, we have:

-6 < x < 4