Express the following in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0.$
$0.5 \overline{7}$

(N/A) Let $x = 0.5 \overline{7}.$
$\therefore x = 0.5777...$ $(1)$
Multiply both sides by $10$ to shift the decimal point before the repeating part:
$10x = 5.777...$ $(2)$
Now,multiply equation $(2)$ by $10$ to shift the decimal point after one repeating digit:
$100x = 57.777...$ $(3)$
Subtract equation $(2)$ from equation $(3)$:
$100x - 10x = 57.777... - 5.777...$
$90x = 52$
$x = \frac{52}{90}$
Simplify the fraction by dividing both numerator and denominator by $2$:
$x = \frac{26}{45}$
Thus,$0.5 \overline{7} = \frac{26}{45}.$