Solve:

$x -\dfrac{x - 1}{2} = 1 -\dfrac{x - 2}{3}$

$x -\dfrac{x - 1}{2} = 1 -\dfrac{x - 2}{3}\\[1em] ⇒ \dfrac{x}{1} -\dfrac{x - 1}{2} = 1 -\dfrac{x - 2}{3}\\[1em]$

Since L.C.M. of denominators 1, 2 and 3 = 6, multiply each term with 6 to get:

$⇒ \dfrac{x \times 6}{1} -\dfrac{6(x - 1)}{2} = 1 \times 6 -\dfrac{6(x - 2)}{3}$

⇒ x $\times$ 6 - 3(x - 1) = 6 - 2(x - 2)

⇒ 6x - (3x - 3) = 6 - (2x - 4)

⇒ 6x - 3x + 3 = 6 - 2x + 4

⇒ 3x + 3 = 10 - 2x

⇒ 3x + 2x = 10 - 3

⇒ 5x = 7

⇒ x = $\dfrac{7}{5}$

Hence, the value of x is $\dfrac{7}{5}$.