To solve this expression, we must first simplify the fractions within the parentheses:

1/6 + 1 3/4 = 1/6 + 4/4 + 3/4
= 1/6 + 7/4
= (12/62) + 7/4
= 2/12 + 7/4
= 2/12 + 21/12
= 23/12

Now, let's substitute this back into the original expression:

0.5 - 2 7/8 / 23/12

Next, convert the mixed number to an improper fraction:

2 7/8 = 8*2 + 7 / 8 = 23/8

Substitute back into the expression:

0.5 - 23/8 / 23/12

Now, we'll convert the whole number to a fraction:

0.5 = 1/2

Substitute it in:

1/2 - 23/8 / 23/12

Now, let's simplify the expression:

1/2 - 23/8 = 4/8 - 23/8 = -19/8

Now, we can substitute this back into the expression:

-19/8 / 23/12

When dividing fractions, we multiply by the reciprocal of the second fraction:

-19/8 * 12/23 = -228/184 = -57/46

Therefore, -57/46 is the final answer.