Arrange in ascending order:

$\dfrac{5}{-7}, \dfrac{-9}{14}, \dfrac{-5}{6} \text{ and } \dfrac{7}{-12}$

First write each rational number with a positive denominator, then write each of them with the LCM of the denominators as the common denominator and arrange the numerators in increasing order.

$\Rightarrow \dfrac{5}{-7} = \dfrac{5 \times (-1)}{-7 \times (-1)} = \dfrac{-5}{7} \\[1em] \Rightarrow \dfrac{7}{-12} = \dfrac{7 \times (-1)}{-12 \times (-1)} = \dfrac{-7}{12}$

So, the given rational numbers are $\dfrac{-5}{7}, \dfrac{-9}{14}, \dfrac{-5}{6} \text{ and } \dfrac{-7}{12}$.

Now write each rational number with the LCM of the denominators as the common denominator.

LCM of 7, 14, 6 and 12 = 84

$\Rightarrow \dfrac{-5}{7} = \dfrac{(-5) \times 12}{7 \times 12} = \dfrac{-60}{84} \\[1em] \Rightarrow \dfrac{-9}{14} = \dfrac{(-9) \times 6}{14 \times 6} = \dfrac{-54}{84} \\[1em] \Rightarrow \dfrac{-5}{6} = \dfrac{(-5) \times 14}{6 \times 14} = \dfrac{-70}{84} \\[1em] \Rightarrow \dfrac{-7}{12} = \dfrac{(-7) \times 7}{12 \times 7} = \dfrac{-49}{84}$

As -70 < -60 < -54 < -49,

So $\dfrac{-70}{84} \lt \dfrac{-60}{84} \lt \dfrac{-54}{84} \lt \dfrac{-49}{84}$

$\Rightarrow \dfrac{-5}{6} \lt \dfrac{-5}{7} \lt \dfrac{-9}{14} \lt \dfrac{-7}{12}$

Hence, the given rational numbers in ascending order are $\mathbf{\dfrac{-5}{6}, \dfrac{5}{-7}, \dfrac{-9}{14} \text{ and } \dfrac{7}{-12}}$.