Solve the following inequation and represent the solution set on the number line.

-3 < $-\dfrac{1}{2} - \dfrac{2x}{3} \le \dfrac{5}{6}$, x ∈ R.

Given,

-3 < $-\dfrac{1}{2} - \dfrac{2x}{3} \le \dfrac{5}{6}$

Solving L.H.S. of the equation,

$\Rightarrow -3 \lt -\dfrac{1}{2} - \dfrac{2x}{3} \\[1em] \Rightarrow \dfrac{2x}{3} \lt -\dfrac{1}{2} + 3 \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{-1 + 6}{2} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{5}{2} \\[1em] \Rightarrow x \lt \dfrac{5}{2} \times \dfrac{3}{2} \\[1em] \Rightarrow x \lt \dfrac{15}{4} \\[1em] \Rightarrow x \lt 3.75 \space ……..(i)$

Solving R.H.S. of the equation,

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

From (i) and (ii) we get,

-2 ≤ x < 3.75

∴ Solution set = {x : -2 ≤ x < 3.75, x ∈ R}.

Solution on the number line is :