Solve the following inequation and represent the solution set on a number line :

$\dfrac{3x}{5} + 2 \lt x + 4 \le \dfrac{x}{2} + 5$; x ∈ R.

Solving L.H.S. of the equation :

$\Rightarrow \dfrac{3x}{5} + 2 \lt x + 4 \\[1em] \Rightarrow \dfrac{3x + 10}{5} \lt x + 4 \\[1em] \Rightarrow 3x + 10 \lt 5(x + 4) \\[1em] \Rightarrow 3x + 10 \lt 5x + 20 \\[1em] \Rightarrow 5x - 3x \gt 10 - 20 \\[1em] \Rightarrow 2x \gt -10 \\[1em] \Rightarrow x \gt -\dfrac{10}{2} \\[1em] \Rightarrow x \gt -5 \text{ ……..(1)}$

Solving R.H.S. of the equation :

$\Rightarrow x + 4 \le \dfrac{x}{2} + 5 \\[1em] \Rightarrow x + 4 \le \dfrac{x + 10}{2} \\[1em] \Rightarrow 2(x + 4) \le x + 10 \\[1em] \Rightarrow 2x + 8 \le x + 10 \\[1em] \Rightarrow 2x - x \le 10 - 8 \\[1em] \Rightarrow x \le 2 \text{ ……..(2)}$

From equation (1) and (2), we get :

-5 < x ≤ 2

Hence, solution set = {x : -5 < x ≤ 2, x ∈ R}.