Find the values of x, which satisfy the inequation $-2\dfrac{5}{6} \lt \dfrac{1}{2} - \dfrac{2x}{3} \le 2$, x ∈ W. Graph the solution set on the number line.

Given,

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

Solving L.H.S. of the inequation,

$\Rightarrow -2\dfrac{5}{6} \lt \dfrac{1}{2} - \dfrac{2x}{3} \\[1em] \Rightarrow -\dfrac{17}{6} \lt \dfrac{1}{2} - \dfrac{2x}{3} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{1}{2} + \dfrac{17}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{3 + 17}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{20}{6} \\[1em] \Rightarrow x \lt \dfrac{20}{6} \times \dfrac{3}{2} \\[1em] \Rightarrow x \lt 5 …….(i)$

Solving R.H.S. of the inequation,

$\Rightarrow \dfrac{1}{2} - \dfrac{2x}{3} \le 2 \\[1em] \Rightarrow \dfrac{2x}{3} \ge \dfrac{1}{2} - 2 \\[1em] \Rightarrow \dfrac{2x}{3} \ge -\dfrac{3}{2} \\[1em] \Rightarrow x \ge -\dfrac{3}{2} \times \dfrac{3}{2} \\[1em] \Rightarrow x \ge -\dfrac{9}{4} \\[1em] \Rightarrow x \ge -2.25 ……..(ii)$

From (i) and (ii) we get,

-2.25 ≤ x < 5

Since, x ∈ W,

∴ Solution set = {0, 1, 2, 3, 4}.

Solution on the number line is :