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

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

Given,

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

Solving L.H.S. of the inequation,

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

Solving R.H.S. of the inequation,

$\Rightarrow \dfrac{3x}{5} - 2 \le\dfrac{-2}{5} + x \\[1em] \Rightarrow \dfrac{3x}{5}-x \le\dfrac{-2}{5} + 2 \\[1em] \Rightarrow \dfrac{3x - 5x }{5} \le \dfrac{-2 + 10}{5} \\[1em] \Rightarrow 3x - 5x \le -2 + 10 \\[1em] \Rightarrow -2x \le 8 \\[1em]$

Dividing by -2 on both sides we get,

⇒ x ≥ -4 (As on dividing by negative number the sign reverses.) ………….(2)

From (1) and (2) we get,

⇒ -4 ≤ x < 5

Since, x ∈ R

Hence, solution set = {x : -4 ≤ x < 5, x ∈ R}.

Solution on the number line is :