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

4x - 19 < $\dfrac{3x}{5} - 2 \le -\dfrac{2}{5} + x$, x ∈ R

Given,

4x - 19 < $\dfrac{3x}{5} - 2 \le -\dfrac{2}{5} + x$

Solving L.H.S. of the equation,

$\Rightarrow 4x - 19 \lt \dfrac{3x}{5} - 2 \\[1em] \Rightarrow 4x - 19 \lt \dfrac{3x - 10}{5} \\[1em] \Rightarrow 5(4x - 19) \lt 3x - 10 \\[1em] \Rightarrow 20x - 95 \lt 3x - 10 \\[1em] \Rightarrow 20x - 3x \lt 95 - 10 \\[1em] \Rightarrow 17x \lt 85 \\[1em] \Rightarrow x \lt 5 ……..(i)$

Solving R.H.S. of the equation,

$\Rightarrow \dfrac{3x}{5} - 2 \le -\dfrac{2}{5} + x \\[1em] \Rightarrow x - \dfrac{3x}{5} \ge -2 + \dfrac{2}{5} \\[1em] \Rightarrow \dfrac{5x - 3x}{5} \ge \dfrac{-10 + 2}{5} \\[1em] \Rightarrow \dfrac{2x}{5} \ge \dfrac{-8}{5} \\[1em] \Rightarrow x \ge -\dfrac{8}{5} \times \dfrac{5}{2} \\[1em] \Rightarrow x \ge -4 ……..(ii)$

From (i) and (ii) we get,

-4 ≤ x < 5

∴ Solution set = {x : -4 ≤ x < 5, x ∈ R}.

Solution on the number line is :