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

$-8\dfrac{1}{2} \lt -\dfrac{1}{2} - 4x \le 7\dfrac{1}{2}$, x ∈ I

Given,

$\Rightarrow -8\dfrac{1}{2} \lt -\dfrac{1}{2} - 4x \le 7\dfrac{1}{2} \\[1em] \Rightarrow -\dfrac{17}{2} \lt -\dfrac{1}{2} - 4x \le \dfrac{15}{2}$

Solving L.H.S. of the inequation,

$\Rightarrow -\dfrac{17}{2} \lt -\dfrac{1}{2} - 4x \\[1em] \Rightarrow 4x\lt \dfrac{17}{2} - \dfrac{1}{2} \\[1em] \Rightarrow 4x\lt \dfrac{16}{2} \\[1em] \Rightarrow 4x \lt 8 \\[1em] \Rightarrow x \lt \dfrac{8}{4} \\[1em] \Rightarrow x \lt 2 \text{ ………..(1)}$

Solving R.H.S. of the inequation,

$\Rightarrow -\dfrac{1}{2} - 4x \le \dfrac{15}{2}\\[1em] \Rightarrow -4x \le \dfrac{15}{2}+\dfrac{1}{2}\\[1em] \Rightarrow -4x \le \dfrac{16}{2}\\[1em] \Rightarrow -4x \le 8$

Dividing by -4 on both sides we get,

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

⇒ x ≥ -2 …..(2)

From (1) and (2) we get,

-2 ≤ x < 2

Since, x ∈ I

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

Solution on the number line is :