Solve the inequation $\;\;$ $\dfrac{-x}{3} \leq \dfrac{x}{2} - \dfrac{4}{3} < \dfrac{1}{6}, \; x \in R$
Write the solution set and represent it on a number line.
Consider $\;$ $\dfrac{-x}{3} \leq \dfrac{x}{2} - \dfrac{4}{3}$
i.e. $\;$ $\dfrac{4}{3} \leq \dfrac{x}{2} + \dfrac{x}{3}$
i.e. $\;$ $\dfrac{4}{3} \leq \dfrac{5x}{6}$
i.e. $\;$ $8 \leq 5x$
i.e. $\;$ $\dfrac{8}{5} \leq x$
i.e. $\;$ $1.6 \leq x$ $\;\;\; \cdots \; (1)$
Consider $\;$ $\dfrac{x}{2} - \dfrac{4}{3} < \dfrac{1}{6}$
i.e. $\;$ $\dfrac{x}{2} < \dfrac{1}{6} + \dfrac{4}{3}$
i.e. $\;$ $\dfrac{x}{2} < \dfrac{9}{6}$
i.e. $\;$ $x < 3$ $\;\;\; \cdots \; (2)$
$\therefore \;$ From equations $(1)$ and $(2)$, the solution set of the given inequation is
$\left\{x \; \bigg | \; 1.6 \leq x < 3, \; \; x \in R \right\}$
