Solve the inequation given below. 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,

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

Solving L.H.S. of the inequation,

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

Solving R.H.S. of the inequation,

$\Rightarrow \dfrac{x}{2} - 1\dfrac{1}{3} \lt \dfrac{1}{6} \\[1em] \Rightarrow \dfrac{x}{2} - \dfrac{4}{3} \lt \dfrac{1}{6} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{1}{6}+ \dfrac{4}{3} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{1 + 8}{6} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{9}{6} \\[1em]$

\Rightarrow x \lt \dfrac{9}{6} \times 2 \\[1em] \Rightarrow x \lt 3 \text{ ……(2)}⇒2x​−131​<61​⇒2x​−34​<61​⇒2x​<61​+34​⇒2x​<61+8​⇒2x​<69​⇒x<69​×2⇒x<3 ……(2)

From (1) and (2) we get,

⇒ 1.6 ≤ x < 3

Since, x ∈ R

Hence, solution set = {x : 1.6 ≤ x < 3, x ∈ R}.

Solution on the number line is :