To find the solution to this expression, we need to perform the operations one by one following the order of operations $PEMDAS$.

6 + 1/2 - 1 1/3 + 7 3/5 - 1 7/8 + 4 8/9

Step 1: 1/2 - 1 1/3 can be simplified by converting the mixed number into an improper fraction:
1/2 - 4/3
= 3/6 - 4/3
= $3-8$/6
= -5/6

Step 2: 7 3/5 - 1 7/8 can be simplified by converting the mixed number into an improper fraction:
7 3/5 - 15/8
= 35/5 + 3/5 - 15/8
= 38/5 - 15/8
To find a common denominator, we multiply the denominators together: 5 8 = 40
= $38</em>8-15\ast 5$ / 40
= $304-75$ / 40
= 229/40

Step 3: Finally, add all the results obtained together:
6 + $-5/6$ + 229/40 + 4 8/9
= 36/6 - 5/6 + 229/40 + 44/9
= $36-5+69+240$ / 40
= 340/40
= 17/2
= 8.5

Therefore, the solution to the expression is 8.5.