Solve the following inequation and write the solution set and represent it on the number line :

$-\dfrac{x}{3} \le \dfrac{x}{2} - 1\dfrac{1}{3} \lt \dfrac{1}{6}$, x ∈ R

Given,

$-\dfrac{x}{3} \le \dfrac{x}{2} - \dfrac{4}{3} \lt \dfrac{1}{6}$

Solving L.H.S. of the inequation,

$\Rightarrow -\dfrac{x}{3} \le \dfrac{x}{2} - \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{x}{2} + \dfrac{x}{3} \ge \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{3x + 2x}{6} \ge \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{5x}{6} \ge \dfrac{4}{3} \\[1em] \Rightarrow x \ge \dfrac{4}{3} \times \dfrac{6}{5} \\[1em] \Rightarrow x \ge \dfrac{8}{5} \\[1em] \Rightarrow x \ge 1.6 …….(i)$

Solving R.H.S. of the inequation,

$\Rightarrow \dfrac{x}{2} - \dfrac{4}{3} \lt \dfrac{1}{6} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{1}{6} + \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{1 + 8}{6} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{9}{6} \\[1em] \Rightarrow x \lt \dfrac{18}{6} \\[1em] \Rightarrow x \lt 3 ……..(ii)$

From (i) and (ii) we get,

1.6 ≤ x < 3

∴ Solution set = {x : 1.6 ≤ x < 3, x ∈ R}

Solution on the number line is :