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

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

Solving L.H.S. of the inequation,

$\Rightarrow -2 \le \dfrac{1}{2} - \dfrac{2x}{3}\\[1em] \Rightarrow \dfrac{1}{2} - \dfrac{2x}{3} \ge –2 \\[1em] \Rightarrow -\dfrac{2x}{3} \ge - 2 -\dfrac{1}{2} \\[1em] \Rightarrow -\dfrac{2x}{3} \ge -\dfrac{5}{2} \\[1em]$

Multiplying by -6 on both sides we get,

⇒ 4x ≤ 15 (As on multiplying by negative number the sign reverses.)

⇒ x ≤ $\dfrac{15}{4}$

⇒ x ≤ 3.75 ………….(1)

Solving R.H.S. of the inequation,

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

Dividing both sides by -2, we get :

⇒ x > -2 (As on dividing by negative number the sign reverses.) ……(2)

From (1) and (2) we get,

-2 > x ≤ 3.75

Since, x ∈ I

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

Solution on the number line is :