To solve the equation, first, we need to convert the mixed numbers into improper fractions.

For 6 3/4:
6 x 4 = 24 + 3 = 27
So, 6 3/4 can be written as 27/4.

For 4 5/7:
4 x 7 = 28 + 5 = 33
So, 4 5/7 can be written as 33/7.

Now, the equation becomes:
2(x - 1) = 27/4 + 33/7

Next, simplify each side of the equation:
2x - 2 = 27/4 + 33/7

To add the fractions on the right side, we need to find a common denominator, which is 28:

2x - 2 = 189/28 + 132/28
2x - 2 = 321/28

Now, we can simplify the right side of the equation further:
2x - 2 = 321/28

To solve for x, first, we can add 2 to both sides of the equation:
2x = 321/28 + 2

Next, write 2 as a fraction with a denominator of 28:
2x = 321/28 + 56/28
2x = 377/28

Finally, divide both sides by 2 to solve for x:
x = 377/28 ÷ 2
x = 377/56

Therefore, the solution to the equation is x = 377/56.