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

Given,

-3 + x ≤ $\frac{7x}{2} + 2$ < 8 + 2x

Solving L.H.S. of the above inequation :

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

Solving R.H.S. of the above inequation :

$$\begin{aligned}\Rightarrow \frac{7x}{2} + 2 \lt 8 + 2x \\[1em] \Rightarrow \frac{7x}{2} - 2x \lt 8 - 2 \\[1em] \Rightarrow \frac{7x - 4x}{2} \lt 6 \\[1em] \Rightarrow \frac{3x}{2} \lt 6 \\[1em] \Rightarrow x \lt \frac{6 \times 2}{3} \\[1em] \Rightarrow x \lt 4 \text{ ..........(2)}\end{aligned}$$

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

-2 ≤ x < 4.

Since, x ∈ I.

x = {-2, -1, 0, 1, 2, 3}.

Hence, solution set = {-2, -1, 0, 1, 2, 3}.