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

$3 \ge \dfrac{x−4}{2} + \dfrac{x}{3} \ge 2$

Given,

$3 \ge \dfrac{x−4}{2}+ \dfrac{x}{3} \ge 2\\[2em] \text{ Solving left side: }\\[1em] 3 \ge \dfrac{x−4}{2}+ \dfrac{x}{3} \\[0.5em] \Rightarrow 3 \ge + \dfrac{3x-12+2x}{6}\\[0.5em] \Rightarrow 3 \ge + \dfrac{5x-12}{6}\\[0.5em] \Rightarrow 18 \ge 5x-12\\[0.5em] \Rightarrow 5x-12 \le 18\\[0.5em] \Rightarrow 5x \le 30\\[0.5em] \Rightarrow x \le 6\\[1em] \text{Solving right side:} \\[1em] \dfrac{x−4}{2} + \dfrac{x}{3} ≥ 2\\[0.5em] \Rightarrow \dfrac{3x-12+2x}{6} \ge 2 \\[0.5em] \Rightarrow \dfrac{5x-12}{6} \ge 2\\[0.5em] \Rightarrow 5x-12 \ge 12 \\[0.5em] \Rightarrow 5x \ge 24\\[0.5em] \Rightarrow x \ge \dfrac{24}{5}\\[0.5em] \Rightarrow x \ge 4\dfrac{4}{5} \\[0.5em]$

∴ Solution Set = {5, 6}.

The graph of the solution set is shown by thick dots on the number line.