First, let's distribute the 1/7 on both terms in the left side of the equation:

1/7(x-14) - (x-1) = 1/7(x+2)
(x/7 - 14/7) - (x-1) = x/7 + 2/7

Now, let's simplify the equation further:

x/7 - 14/7 - x + 1 = x/7 + 2/7

Next, let's combine like terms:

-x/7 - 13/7 = x/7 + 2/7

Now, let's isolate x on one side of the equation:

-x/7 - 13/7 - x/7 = 2/7

-x/7 - x/7 = 2/7 + 13/7

-2x/7 = 15/7

Now, let's solve for x:

2x = -15

x = -15/2

Therefore, the solution to the equation 1/7(x-14)-(x-1)=1/7(x+2) is x = -15/2.