Find the values of x, which satisfy the inequation :

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

Graph the solution set on the real number line.

Solving L.H.S. of the above inequation :

$\Rightarrow -2 \le \dfrac{1}{2} - \dfrac{2x}{3} \\[1em] \Rightarrow \dfrac{2x}{3} \le \dfrac{1}{2} + 2 \\[1em] \Rightarrow \dfrac{2x}{3} \le \dfrac{5}{2} \\[1em] \Rightarrow x \le \dfrac{5}{2} \times \dfrac{3}{2} \\[1em] \Rightarrow x \le \dfrac{15}{4} \\[1em] \Rightarrow x \le 3\dfrac{3}{4}\text{ ………..(1)}$

Solving R.H.S. of the inequation :

$\Rightarrow \dfrac{1}{2} - \dfrac{2x}{3} \lt 1\dfrac{5}{6} \\[1em] \Rightarrow \dfrac{1}{2} - \dfrac{2x}{3} \lt \dfrac{11}{6} \\[1em] \Rightarrow \dfrac{1}{2} - \dfrac{11}{6} \lt \dfrac{2x}{3} \\[1em] \Rightarrow \dfrac{2x}{3} \gt \dfrac{3 - 11}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \gt -\dfrac{8}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \gt -\dfrac{4}{3} \\[1em] \Rightarrow x \gt -\dfrac{4}{3} \times \dfrac{3}{2} \\[1em] \Rightarrow x \gt -2 \text{ ………(2)}$

From (1) and (2),

-2 < x ≤ $3\dfrac{3}{4}$

Since, x ∈ N

∴ x = {1, 2, 3}.

Hence, x = {1, 2, 3}.