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

Given, inequation : 5x - 21 < $\frac{5x}{7} - 6 \le -3\frac{3}{7} + x$

Solving L.H.S. of the inequation :

$$\begin{aligned}\Rightarrow 5x - 21 \lt \frac{5x}{7} - 6\\[1em] \Rightarrow 5x - \frac{5x}{7} \lt 21 - 6 \\[1em] \Rightarrow \frac{35x - 5x}{7} \lt 15 \\[1em] \Rightarrow \frac{30x}{7} \lt 15 \\[1em] \Rightarrow x \lt \frac{7 \times 15}{30} \\[1em] \Rightarrow x \lt \frac{7}{2} \text{ ..........(1)}\end{aligned}$$

Solving R.H.S. of the inequation :

$$\begin{aligned}\Rightarrow \frac{5x}{7} - 6 \le -3\frac{3}{7} + x \\[1em] \Rightarrow \frac{5x}{7} - 6 \le -\frac{24}{7} + x \\[1em] \Rightarrow x - \frac{5x}{7} \ge -6 + \frac{24}{7} \\[1em] \Rightarrow \frac{7x - 5x}{7} \ge \frac{-42 + 24}{7} \\[1em] \Rightarrow \frac{2x}{7} \ge \frac{-18}{7} \\[1em] \Rightarrow 2x \ge -18 \\[1em] \Rightarrow x \ge \frac{-18}{2} \\[1em] \Rightarrow x \ge -9 \text{ ..........(2)}\end{aligned}$$

From equation (1) and (2),

Solution set = {x : -9 ≤ x < $\frac{7}{2}$, x ∈ R}

Hence, solution set = {x : -9 ≤ x < $\frac{7}{2}$, x ∈ R}.