Find the solution set of each of the following inequations :

(i) 2 < x - 3 < 7, x ∈ N

(ii) 10 < 4x - 5 < 21, x ∈ N

(iii) 2 - x < 4x - 7 < 11 - 2x, x ∈ Z

(iv) 4 - 2x < 3x + 19 < 42 - 5x, x ∈ Z

(v) -5 < $\dfrac{x}{2}$ - 3 < $\dfrac{5}{2}$, x ∈ Z

(vi) 9 - $\dfrac{2}{3}$x < 5x - 11 < 17 - $\dfrac{x}{4}$, x ∈ Z

(i) 2 < x - 3 < 7, x ∈ N

We have:

2 < x - 3 < 7

Let's separate the inequalities:

Case 1:

2 < x - 3

⇒ 2 + 3 < x $\quad$ [Adding 3 on both sides]

⇒ 5 < x

⇒ x > 5

Natural numbers greater than 5 are {6, 7, 8, 9, 10, …}

∴ Solution set A = {6, 7, 8, 9, 10, …}

Case 2:

x - 3 < 7

⇒ x < 7 + 3 $\quad$ [Adding 3 on both sides]

⇒ x < 10

Natural numbers less than 10 are {1, 2, 3, 4, 5, 6, 7, 8, 9}

∴ Solution set B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Final solution set = A ∩ B = {6, 7, 8, 9}

(ii) 10 < 4x - 5 < 21, x ∈ N

We have:

10 < 4x - 5 < 21

Let's separate the inequalities:

Case 1:

10 < 4x - 5

⇒ 10 + 5 < 4x $\quad$ [Adding 5 on both sides]

⇒ 15 < 4x

⇒ $\dfrac{15}{4}$ < x $\quad$ [Dividing both sides by 4]

⇒ 3.75 < x

⇒ x > 3.75

Natural numbers greater than 3.75 are {4, 5, 6, 7, …}

∴ Solution set A = {4, 5, 6, 7, …}

Case 2:

4x - 5 < 21

⇒ 4x < 21 + 5 $\quad$ [Adding 5 on both sides]

⇒ 4x < 26

⇒ x < $\dfrac{26}{4} \quad \text{[Dividing both sides by 4]}$

⇒ x < 6.5

Natural numbers less than 6.5 are {1, 2, 3, 4, 5, 6}

∴ Solution set B = {1, 2, 3, 4, 5, 6}

Final solution set = A ∩ B = {4, 5, 6}

(iii) 2 - x < 4x - 7 < 11 - 2x, x ∈ Z

We have:

2 - x < 4x - 7 < 11 - 2x

Let's separate the inequalities:

Case 1:

2 - x < 4x - 7

⇒ 2 + 7 < 4x + x $\quad$ [Adding 7 and x on both sides]

⇒ 9 < 5x

⇒ $\dfrac{9}{5}$ < x $\quad$ [Dividing both sides by 5]

⇒ 1.8 < x

⇒ x > 1.8

Integers greater than 1.8 are {2, 3, 4, 5, …}

∴ Solution set A = {2, 3, 4, 5, …}

Case 2:

4x - 7 < 11 - 2x

⇒ 4x + 2x < 11 + 7 $\quad$ [Adding 7 and 2x on both sides]

⇒ 6x < 18

⇒ x < $\dfrac{18}{6} \quad \text{[Dividing both sides by 6]}$

⇒ x < 3

Integers less than 3 are {…, -1, 0, 1, 2}

∴ Solution set B = {…, -1, 0, 1, 2}

Final solution set = A ∩ B = {2}

(iv) 4 - 2x < 3x + 19 < 42 - 5x, x ∈ Z

We have:

4 - 2x < 3x + 19 < 42 - 5x

Let's separate the inequalities:

Case 1:

4 - 2x < 3x + 19

⇒ 4 - 19 < 3x + 2x $\quad$ [Subtracting 19 and adding 2x on both sides]

⇒ -15 < 5x

⇒ $\dfrac{-15}{5}$ < x $\quad$ [Dividing both sides by 5]

⇒ -3 < x

⇒ x > -3

Integers greater than -3 are {-2, -1, 0, 1, 2, …}

∴ Solution set A = {-2, -1, 0, 1, 2, …}

Case 2:

3x + 19 < 42 - 5x

⇒ 3x + 5x < 42 - 19 $\quad$ [Subtracting 19 and adding 5x on both sides]

⇒ 8x < 23

⇒ x < $\dfrac{23}{8} \quad \text{[Dividing both sides by 8]}$

⇒ x < 2.875

Integers less than 2.875 are {…, -2, -1, 0, 1, 2}

∴ Solution set B = {…, -2, -1, 0, 1, 2}

Final solution set = A ∩ B = {-2, -1, 0, 1, 2}

(v) -5 < $\dfrac{x}{2}$ - 3 < $\dfrac{5}{2}$, x ∈ Z

We have:

-5 < $\dfrac{x}{2}$ - 3 < $\dfrac{5}{2}$

Let's separate the inequalities:

Case 1:

-5 < $\dfrac{x}{2}$ - 3

⇒ -5 + 3 < $\dfrac{x}{2} \quad \text{[Adding 3 on both sides]}$

⇒ -2 < $\dfrac{x}{2}$

⇒ -2 x 2 < x $\quad$ [Multiplying 2 on both sides]

⇒ -4 < x

⇒ x > -4

Integers greater than -4 are {-3, -2, -1, 0, …}

∴ Solution set A = {-3, -2, -1, 0, …}

Case 2:

$\phantom{=} \dfrac{x}{2} - 3 \lt \dfrac{5}{2} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{5}{2} + 3 \quad \text{[Adding 3 on both sides]} \\[1em] \Rightarrow \dfrac{x}{2} \lt \dfrac{5 + 6}{2} \quad \text{[Since both denominators are same, we cancel them]} \\[1em] \Rightarrow x \lt 5 + 6 \\[1em] \Rightarrow x \lt 11$

Integers less than 11 are {…, 8, 9, 10}

∴ Solution set B = {…, 8, 9, 10}

Final solution set = A ∩ B = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vi) 9 - $\dfrac{2}{3}$x < 5x - 11 < 17 - $\dfrac{x}{4}$, x ∈ Z

We have:

9 - $\dfrac{2}{3}$x < 5x - 11 < 17 - $\dfrac{x}{4}$

Let's separate the inequalities:

Case 1:

$\phantom{=} 9 - \dfrac{2}{3}x \lt 5x - 11 \\[1em] \Rightarrow 9 + 11 \lt 5x + \dfrac{2}{3}x \quad \text{[Adding 11 on both sides]} \\[1em] \Rightarrow 20 \lt \dfrac{15x + 2x}{3} \\[1em] \Rightarrow 20 \lt \dfrac{17x}{3} \\[1em] \Rightarrow 20 \times 3 \lt 17x \quad \text{[Multiplying 3 on both sides]} \\[1em] \Rightarrow 60 \lt 17x \\[1em] \Rightarrow \dfrac{60}{17} \lt x \quad \text{[Dividing 17 on both sides]} \\[1em] \Rightarrow 3.52.. \lt x \\[1em] \Rightarrow x \gt 3.52..$

Integers greater than 3.52.. are {4, 5, 6, …}

∴ Solution set A = {4, 5, 6, …}

Case 2:

$\phantom{=} 5x - 11 \lt 17 - \dfrac{x}{4} \\[1em] \Rightarrow 5x + \dfrac{x}{4} \lt 17 + 11 \quad \text{[Adding 11 and} \dfrac{x}{4} \text{ on both sides]} \\[1em] \Rightarrow \dfrac{20x + x}{4} \lt 28 \\[1em] \Rightarrow \dfrac{21x}{4} \lt 28 \\[1em] \Rightarrow 21x \lt 28 \times 4 \quad \text{[Multiplying 4 on both sides]} \\[1em] \Rightarrow 21x \lt 112 \\[1em] \Rightarrow x \lt \dfrac{112}{21} \quad \text{[Dividing 21 on both sides]} \\[1em] \Rightarrow x \lt 5.33..$

Integers less than 5.33.. are {…, 3, 4, 5}

∴ Solution set B = {…, 3, 4, 5}

Final solution set = A ∩ B = {4, 5}