Solve the following inequation, write the solution set and represent it on the real number line.

Solving L.H.S of the equation $2x - \frac{5}{3} \lt \frac{3x}{5} + 10 \le \frac{4x}{5} + 11 ;\ x \in R$ we get,

$$\begin{aligned}\Rightarrow 2x - \frac{5}{3} \lt \frac{3x}{5} + 10 \\[1em] \Rightarrow \frac{6x - 5}{3} \lt \frac{3x + 50}{5} \\[1em] \text{Multiplying both side by 15, we get:} \\[1em] \Rightarrow 15\Big(\frac{6x - 5}{3}\Big) \lt 15\Big(\frac{3x + 50}{5}\Big) \\[1em] \Rightarrow 5(6x - 5) \lt 3(3x + 50) \\[1em] \Rightarrow 30x - 25 \lt 9x + 150 \\[1em] \Rightarrow 30x - 9x \lt 25 + 150 \\[1em] \Rightarrow 21x \lt 175 \\[1em] \Rightarrow x \lt \frac{175}{21} \\[1em] \Rightarrow x \lt \frac{25}{3} \text{ ........(1)}\end{aligned}$$

Solving R.H.S of the equation $2x - \frac{5}{3} \lt \frac{3x}{5} + 10 \le \frac{4x}{5} + 11 ;\ x \in R$ we get,

$$\begin{aligned}\Rightarrow \frac{3x}{5} + 10 \le \frac{4x}{5} + 11 \\[1em] \Rightarrow \frac{3x + 50}{5} \le \frac{4x + 55}{5} \\[1em] \text{Multiplying both side by 5, we get:} \\[1em] \Rightarrow 5\Big(\frac{3x + 50}{5}\Big) \le 5\Big(\frac{4x + 55}{5}\Big) \\[1em] \Rightarrow 3x + 50 \le 4x + 55 \\[1em] \Rightarrow 50 - 55 \le 4x - 3x \\[1em] \Rightarrow -5 \le x \\[1em] \Rightarrow x \ge -5 \text{ ........(2)}\end{aligned}$$

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

-5 ≤ x < $\frac{25}{3}$.

Hence, solution set equals to -5 ≤ x < $\frac{25}{3}$, x ∈ R.