Given that x ∈ I, solve the inequation and graph the solution on number line :

3 ≥ $\dfrac{x - 4}{2} + \dfrac{x}{3} \ge 2$

Given,

3 ≥ $\dfrac{x - 4}{2} + \dfrac{x}{3} \ge 2$

Solving L.H.S. of the inequation,

$\Rightarrow 3 \ge \dfrac{x - 4}{2} + \dfrac{x}{3} \\[1em] \Rightarrow \dfrac{3(x - 4) + 2x}{6} \le 3 \\[1em] \Rightarrow 3x - 12 + 2x \le 18 \\[1em] \Rightarrow 5x \le 18 + 12 \\[1em] \Rightarrow 5x \le 30 \\[1em] \Rightarrow x \le 6 ……..(i)$

Solving R.H.S. of the inequation,

$\Rightarrow \dfrac{x - 4}{2} + \dfrac{x}{3} \ge 2 \\[1em] \dfrac{3(x - 4) + 2x}{6} \ge 2 \\[1em] 3x - 12 + 2x \ge 12 \\[1em] 5x \ge 24 \\[1em] x \ge \dfrac{24}{5} …….(ii)$

From (i) and (ii) we get,

$\dfrac{24}{5} \le x \le 6$

∴ Solution set = {5, 6}.

Solution on the number line is :