What should be subtracted from $\dfrac{-3}{4}$ to get $\dfrac{-5}{6}$

Let the required number be x. Then,

$\dfrac{-3}{4} - x = \dfrac{-5}{6} \\[1em] \Rightarrow \dfrac{-3}{4} = \dfrac{-5}{6} + x \\[1em] \Rightarrow x = \dfrac{-3}{4} - \left(\dfrac{-5}{6}\right) \\[1em] = \dfrac{-3}{4} + \left(\text{additive inverse of } \dfrac{-5}{6}\right) \\[1em] x = \dfrac{-3}{4} + \dfrac{5}{6}$

L.C.M. of denominators 4 and 6 is 12.

Now, expressing each fraction with denominator 12:

$\dfrac{-3 \times 3}{4 \times 3} + \dfrac{5 \times 2}{6 \times 2} \\[1em] = \dfrac{-9}{12} + \dfrac{10}{12} \\[1em] = \dfrac{-9 + 10}{12} \\[1em] = \dfrac{1}{12}$

The required number is $\dfrac{1}{12}$.