Solve the inequation $\;$ $\dfrac{-1}{3} \leq \dfrac{x}{2} - 1\dfrac{1}{3} < \dfrac{1}{6}, \; x \in R$ $\;$ and represent the solution set on a number line.
Consider $\;$ $\dfrac{-1}{3} \leq \dfrac{x}{2} - 1\dfrac{1}{3}$
i.e. $\;$ $\dfrac{-1}{3} \leq \dfrac{x}{2} - \dfrac{4}{3}$
i.e. $\;$ $\dfrac{4}{3} - \dfrac{1}{3} \leq \dfrac{x}{2}$
i.e. $\;$ $1 \leq \dfrac{x}{2}$
i.e. $\;$ $2 \leq x$ $\;\;\; \cdots \; (1)$
Consider $\;$ $\dfrac{x}{2} - 1\dfrac{1}{3} < \dfrac{1}{6}$
i.e. $\;$ $\dfrac{x}{2} - \dfrac{4}{3} < \dfrac{1}{6}$
i.e. $\;$ $\dfrac{x}{2} < \dfrac{1}{6} + \dfrac{4}{3}$
i.e. $\;$ $\dfrac{x}{2} < \dfrac{3}{2}$
i.e. $\;$ $x < 3$ $\;\;\; \cdots \; (2)$
$\therefore \;$ From equations $(1)$ and $(2)$, the solution set of the given inequation is
$\left\{x \; \big | \; 2 \leq x < 3, \; x \in R \right\}$
