Find the range of values of x, which satisfy :

$-\dfrac{1}{3} \le \dfrac{x}{2} + 1\dfrac{2}{3} \lt 5\dfrac{1}{6}$

Graph, in each of the following cases, the values of x on different real number lines:

(i) x ∈ W

(ii) x ∈ Z

(iii) x ∈ R

Given,

$\Rightarrow -\dfrac{1}{3} \le \dfrac{x}{2} + 1\dfrac{2}{3} \lt 5\dfrac{1}{6}$

Solving L.H.S of the above equation,

$\Rightarrow -\dfrac{1}{3} \le \dfrac{x}{2} + 1\dfrac{2}{3} \\[1em] \Rightarrow -\dfrac{1}{3} \le \dfrac{x}{2} + \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{x}{2} \ge -\dfrac{1}{3} - \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{x}{2} \ge -\dfrac{6}{3} \\[1em] \Rightarrow \dfrac{x}{2} \ge -2 \\[1em] \Rightarrow x \ge -2 \times 2 \\[1em] \Rightarrow x \ge -4 \space …….(i)$

Solving R.H.S of the above equation,

$\Rightarrow \dfrac{x}{2} + 1\dfrac{2}{3} \lt 5\dfrac{1}{6} \\[1em] \Rightarrow \dfrac{x}{2} + \dfrac{5}{3} \lt \dfrac{31}{6} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{31}{6} - \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{31 - 10}{6} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{21}{6} \\[1em] \Rightarrow x \lt 2 \times \dfrac{21}{6} \\[1em] \Rightarrow x \lt \dfrac{42}{6} \\[1em] \Rightarrow x \lt 7 \space …….(ii)$

From (i) and (ii) we get,

⇒ -4 ≤ x < 7

(i) In this case x ∈ W

∴ Solution set = {0, 1, 2, 3, 4, 5, 6}

Solution on the number line is :

(ii) In this case x ∈ Z

∴ Solution set = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

Solution on the number line is :

(iii) In this case x ∈ R

∴ Solution set = {x : -4 ≤ x < 7, x ∈ R}

Solution on the number line is :