Find the values of x, which satisfy the inequation :

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

Graph the solution on the number line.

Given,

$\Rightarrow -2 \le \dfrac{1}{2} - \dfrac{2x}{3} \le 1\dfrac{5}{6} \\[1em] \Rightarrow -2 \le \dfrac{3 - 4x}{6} \le \dfrac{11}{6} \\[1em]$

Solving L.H.S. of the equation,

$\Rightarrow -2 \le \dfrac{3 - 4x}{6} \\[1em] \Rightarrow -12 \le 3 - 4x \\[1em] \Rightarrow 4x \le 3 + 12 \\[1em] \Rightarrow x \le \dfrac{15}{4} ……..(i)$

Solving R.H.S. of the equation,

$\Rightarrow \dfrac{1}{2} - \dfrac{2x}{3} \le \dfrac{11}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \ge \dfrac{1}{2} - \dfrac{11}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \ge \dfrac{3 - 11}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \ge -\dfrac{8}{6} \\[1em] \Rightarrow x \ge -\dfrac{8}{6} \times \dfrac{3}{2} \\[1em] \Rightarrow x \ge -2 ………(ii)$

From (i) and (ii) we get,

$-2 \le x \le \dfrac{15}{4}$

Since, x ∈ N,

∴ Solution set = {1, 2, 3}.

Solution on the number line is :