To solve this expression, we need to first find a common denominator for all the fractions involved.

The common denominator for 6, 20, and 90 is 180.

So, we rewrite the expression with the common denominator:

5/6 - 712/180 + 9/20 - ... - 19/90

Next, we convert all the fractions to have a denominator of 180:

5/6 = 150/180
712/180 = 712/180
9/20 = 81/180
19/90 = 38/180

Now, we can rewrite the expression:

150/180 - 712/180 + 81/180 - ... - 38/180

Combining all the fractions, we get:

(150 - 712 + 81 - ... - 38) / 180

Now, we can simplify the numerator:

150 - 712 + 81 - ... - 38 = -539

Therefore, the final answer is:

-539 / 180