To solve this expression, we need to follow the order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

First, let's simplify the expression step by step:

1) 1/2 + 2/3
Convert both fractions to have a common denominator. In this case, the common denominator is 6.
1/2 = 3/6
2/3 = 4/6

Add the fractions together:
3/6 + 4/6 = 7/6

2) Now, multiply the result from step 1 by 7 1/7
Convert 7 1/7 to an improper fraction:
7 1/7 = 7 + 1/7
= 49/7 + 1/7
= 50/7

Now, multiply 7/6 by 50/7:
(7/6) (50/7) = (750) / (6*7) = 350 / 42 = 25/3

Therefore, the final result of the expression 1/2 + 2/3 * 7 1/7 is 25/3.