Solve the inequation given below. Write the solution set and represent it on the number line:

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

Given,

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

Solving L.H.S. of the inequation,

$\Rightarrow -2\dfrac{5}{6} \lt \dfrac{1}{2} - \dfrac{2x}{3} \\[1em] \Rightarrow -\dfrac{17}{6} \lt \dfrac{1}{2} - \dfrac{2x}{3} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{1}{2} + \dfrac{17}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{3 + 17}{6} \\[1em] \Rightarrow \dfrac{2x}{3} \lt \dfrac{20}{6} \\[1em] \Rightarrow x \lt \dfrac{20}{6} \times \dfrac{3}{2} \\[1em] \Rightarrow x \lt \dfrac{60}{12} \\[1em] \Rightarrow x \lt 5 \text{ ………(1)}$

Solving R.H.S. of the inequation,

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

\Rightarrow -\dfrac{2x}{3} \le \dfrac{3}{2}⇒21​−32x​≤2⇒−32x​≤2−21​⇒−32x​≤24−1​⇒−32x​≤23​

Multiplying both sides by $-\dfrac{3}{2}$, we get :

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

From (1) and (2) we get,

-2.25 ≤ x < 5,

Since, x ∈ W

Hence, solution set = {0, 1, 2, 3, 4}.

Solution on the number line is :