For each set of rational numbers, given below, verify the associative property of addition of rational numbers:

$\dfrac{-7}{9} , \dfrac{2}{-3}$ and $\dfrac{-5}{18}$

To prove: $\Big(\dfrac{-7}{9} + \dfrac{2}{-3}\Big) + \dfrac{-5}{18} = \dfrac{-7}{9} + \Big(\dfrac{2}{-3} + \dfrac{-5}{18}\Big)$

Taking LHS:

$\Big(\dfrac{-7}{9} + \dfrac{2}{-3}\Big) + \dfrac{-5}{18} \\[1em] = \Big(\dfrac{-7}{9} + \dfrac{-2}{3}\Big) + \dfrac{-5}{18} \\[1em]$ LCM of 9 and 3 is 3 x 3 = 9

$= \Big(\dfrac{-7 \times 1}{9 \times 1} + \dfrac{-2 \times 3}{3 \times 3}\Big) + \dfrac{-5}{18} \\[1em] = \Big(\dfrac{-7}{9} + \dfrac{-6}{9}\Big) + \dfrac{-5}{18} \\[1em] = \Big(\dfrac{-7 + (-6)}{9}\Big) + \dfrac{-5}{18} \\[1em] = \dfrac{-13}{9}+ \dfrac{-5}{18} \\[1em]$

LCM of 9 and 18 is 2 x 9 = 18

$= \dfrac{-13 \times 2}{9 \times 2} + \dfrac{-5 \times 1}{18 \times 1} \\[1em] = \dfrac{-26}{18} + \dfrac{-5}{18} \\[1em] = \dfrac{-26 + (-5)}{18} \\[1em] = \dfrac{-31}{18} \\[1em]$

Taking RHS:

$\dfrac{-7}{9} + \Big(\dfrac{2}{-3} + \dfrac{-5}{18}\Big) \\[1em] = \dfrac{-7}{9} + \Big(\dfrac{-2}{3} + \dfrac{-5}{18}\Big) \\[1em]$

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

$\dfrac{-7}{9} + \Big(\dfrac{-2 \times 6}{3 \times 6} + \dfrac{-5 \times 1}{18 \times1}\Big) \\[1em] = \dfrac{-7}{9} + \Big(\dfrac{-12}{18} + \dfrac{-5}{18}\Big) \\[1em] = \dfrac{-7}{9} + \Big( \dfrac{-12 +(-5)}{18}\Big) \\[1em] = \dfrac{-7}{9} + \dfrac{-17}{18} \\[1em]$

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

$= \dfrac{-7 \times 2}{9 \times 2} + \dfrac{-17 \times 1}{18 \times 1} \\[1em] = \dfrac{-14}{18} + \dfrac{-17}{18} \\[1em] = \dfrac{-14 + (-17)}{18} \\[1em] = \dfrac{-31}{18} \\[1em]$

∴ LHS = RHS

$\Big(\dfrac{-7}{9} + \dfrac{2}{-3}\Big) + \dfrac{-5}{18} = \dfrac{-7}{9} + \Big(\dfrac{2}{-3} + \dfrac{-5}{18}\Big)$

So, the associative property for the addition of the rational number is verified.