To simplify the expression 7 - 2√7 / 2 - √7 + 6/√7 + 1, we can follow these steps:

First, let's simplify the expression 7 - 2√7 / 2 - √7:

To simplify the numerator, multiply both the numerator and denominator by the conjugate of the denominator, which is 2 + √7:

(7 - 2√7)(2 + √7) = 14 + 7√7 - 4√7 - 2(√7)^2
= 14 + 7√7 - 4√7 - 2(7)
= 14 + 7√7 - 4√7 - 14
= 7√7 - 4√7

Now, let's simplify the denominator:

2 - √7

Now, let's simplify the expression 6/√7:

To simplify this, multiply both the numerator and denominator by √7:

(6/√7)(√7/√7) = 6√7/7

Finally, let's simplify the expression 7√7 - 4√7 / 2 - √7 + 6√7/7 + 1:

Combine like terms in the numerator:

(7√7 - 4√7) = 3√7

Combine like terms in the denominator:

(2 - √7) + (6√7/7) + 1 = 2 - √7 + 6√7/7 + 1
= 3 - √7 + 6√7/7

Therefore, the simplified expression is 3√7 / 3 - √7 + 6√7/7.