If x ∈ N, find the solution set of each of the following inequations :

(i) 4x < 13

(ii) 2x - 9 < -1

(iii) 3 - x < -2

(iv) 5 - 7x > - 16

(v) $\dfrac{4}{7}-\dfrac{x}{4}$ > - 2

(vi) $-\dfrac{1}{2}$ > $\dfrac{1}{4} - \dfrac{x}{3}$

(i) 4x < 13

We have:

⇒ 4x < 13

⇒ x < $\dfrac{13}{4} \quad \text{[Dividing both sides by 4]}$

⇒ x < 3.25

Natural numbers less than 3.25 are {1, 2, 3}.

∴ Solution set = {1, 2, 3}

(ii) 2x - 9 < -1

We have:

2x - 9 < -1

⇒ 2x < -1 + 9 $\quad$ [Adding 9 on both sides]

⇒ 2x < 8

⇒ x < $\dfrac{8}{2} \quad \text{[Dividing both sides by 2]}$

⇒ x < 4

Natural numbers less than 4 are {1, 2, 3}.

∴ Solution set = {1, 2, 3}

(iii) 3 - x < -2

We have:

3 - x < -2

⇒ -x < -2 - 3 $\quad$ [Subtracting 3 from both sides]

⇒ -x < -5

⇒ x > 5 $\quad$ [Multiplying -1 on both sides and reversing the sign]

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

∴ Solution set = {6, 7, 8, 9, …}

(iv) 5 - 7x > - 16

We have:

5 - 7x > - 16

⇒ -7x > -16 - 5 $\quad$ [Subtracting 5 from both sides]

⇒ -7x > -21

Dividing by a negative number reverses the inequality:

⇒ x < $\dfrac{-21}{-7} \quad \text{[Dividing both sides by -7]}$

⇒ x < 3

Natural numbers less than 3 are {1, 2}

∴ Solution set = {1, 2}

(v) $\dfrac{4}{7}-\dfrac{x}{4}$ > - 2

We have:

$\phantom{=} \dfrac{4}{7}-\dfrac{x}{4} \gt - 2 \\[1em] \Rightarrow -\dfrac{x}{4} \gt -2 - \dfrac{4}{7} \quad \text{[Subtracting } \dfrac{4}{7} \text{ from both sides]} \\[1em] \Rightarrow -\dfrac{x}{4} \gt \dfrac{-14 - 4}{7} \\[1em] \Rightarrow -\dfrac{x}{4} \gt \dfrac{-18}{7} \\[1em] \Rightarrow -x \gt \dfrac{-18}{7} \times 4 \quad \text{[Multiplying 4 on both sides]} \\[1em] \Rightarrow -x \gt \dfrac{-72}{7} \\[1em] \Rightarrow -x \gt -10.28… \\[1em] \Rightarrow x \lt 10.28… \quad \text{[Multiplying -1 on both sides and reversing the sign]}$

Natural numbers less than 10.28 are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

∴ Solution set = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vi) $-\dfrac{1}{2}$ > $\dfrac{1}{4} - \dfrac{x}{3}$

We have:

$\phantom{=} -\dfrac{1}{2} \gt \dfrac{1}{4} - \dfrac{x}{3} \\[1em] \Rightarrow -\dfrac{1}{2} - \dfrac{1}{4} \gt - \dfrac{x}{3} \quad \text{[Subtracting } \dfrac{1}{4} \text{ from both sides]} \\[1em] \Rightarrow \dfrac{-2 - 1}{4} \gt - \dfrac{x}{3} \\[1em] \Rightarrow \dfrac{-3}{4} \gt - \dfrac{x}{3} \\[1em] \Rightarrow \dfrac{-3}{4} \times 3 \gt -x \quad \text{[Multiplying 3 on both sides]} \\[1em] \Rightarrow \dfrac{-9}{4} \gt -x \\[1em] \Rightarrow -2.25 \gt -x \\[1em] \Rightarrow 2.25 \lt x \quad \text{[Multiplying -1 on both sides and reversing the sign]} \\[1em]$

Natural numbers greater than 2.25 are {3, 4, 5, …}

∴ Solution set = {3, 4, 5, …}