Insert six rational numbers between $\dfrac{5}{6}$ and $\dfrac{8}{9}$.

LCM of 6 and 9 is 2 x 3 x 3 = 18

Make denominator of each given rational number equal to 18 (the LCM).

$\dfrac{5}{6} = \dfrac{5 \times 3}{6 \times 3} = \dfrac{15}{18}$

and

$\dfrac{8}{9} = \dfrac{8 \times 2}{9 \times 2} = \dfrac{16}{18}$

Since six rational numbers are required between $\dfrac{5}{6}$ and $\dfrac{8}{9}$; multiply the numerator and the denominator of each rational number by 6 + 1 = 7.

$\dfrac{15}{18} = \dfrac{15 \times 7}{18 \times 7} = \dfrac{105}{126}$

and

$\dfrac{16}{18} = \dfrac{16 \times 7}{18 \times 7} = \dfrac{112}{126}$

Required rational numbers between $\dfrac{5}{6}$ and $\dfrac{8}{9}$ are : $\dfrac{105}{126} , \dfrac{106}{126} , \dfrac{107}{126} , \dfrac{108}{126} , \dfrac{109}{126} , \dfrac{110}{126} , \dfrac{111}{126} , \dfrac{112}{126}$

= $\dfrac{5}{6} , \dfrac{53}{63} , \dfrac{107}{126} , \dfrac{6}{7} , \dfrac{109}{126} , \dfrac{55}{63} , \dfrac{37}{42} , \dfrac{8}{9}$

Hence, $\dfrac{5}{6} , \dfrac{53}{63} , \dfrac{107}{126} , \dfrac{6}{7} , \dfrac{109}{126}$ and $\dfrac{37}{42}$ lie between $\dfrac{5}{6}$ and $\dfrac{8}{9}$.