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

Given,

$-1 \lt \frac{2x - 3}{3} - \frac{x}{5} \le 1, x \in R$

Solving L.H.S of the inequation,

$$\begin{aligned}\Rightarrow -1 \lt \frac{2x - 3}{3} - \frac{x}{5} \\[1em] \Rightarrow -1 \lt \frac{10x - 15 - 3x}{15} \\[1em] \Rightarrow -1 \lt \frac{7x - 15}{15} \\[1em] \Rightarrow -15 \lt 7x - 15 \\[1em] \Rightarrow -15 + 15 \lt 7x \\[1em] \Rightarrow 0 \lt 7x \\[1em] \Rightarrow x \gt 0.\end{aligned}$$

Solving R.H.S of the inequation,

$$\begin{aligned}\Rightarrow \frac{2x - 3}{3} - \frac{x}{5} \le 1 \\[1em] \Rightarrow \frac{10x - 15 - 3x}{15} \le 1 \\[1em] \Rightarrow \frac{7x - 15}{15} \le 1 \\[1em] \Rightarrow 7x - 15 \le 15 \\[1em] \Rightarrow 7x \le 15 + 15 \\[1em] \Rightarrow 7x \le 30 \\[1em] \Rightarrow x \le \frac{30}{7}.\end{aligned}$$

∴ 0 < x ≤ $\frac{30}{7}$

Solution set = $\Big(0, \frac{30}{7}\Big]$

Hence, solution set = $\Big(0, \frac{30}{7}\Big]$.