If x ∈ Z⁺, find the solution set of each of the following inequations. Represent each solution set on the number line.

(i) 7x < 17

(ii) 4x - 11 < 5

(iii) 8 - x > $\dfrac{1}{3}$

(iv) 4(x + 5) < 29

(v) 5 > $\dfrac{2}{3}$x

(vi) 2 - $\dfrac{7x}{29}$ < $\dfrac{5}{3}$

(i) 7x < 17

We have:

7x < 17

⇒ x < $\dfrac{17}{7} \quad \text{[Dividing both sides by 7]}$

⇒ x < 2.42…

Positive integers less than 2.42… are {1, 2}

∴ Solution set = {1, 2}

(ii) 4x - 11 < 5

We have:

4x - 11 < 5

⇒ 4x < 5 + 11 $\quad$ [Adding 11 on both sides]

⇒ 4x < 16

⇒ x < $\dfrac{16}{4} \quad \text{[Dividing both sides by 4]}$

⇒ x < 4

Positive integers less than 4 are {1, 2, 3}

∴ Solution set = {1, 2, 3}

(iii) 8 - x > $\dfrac{1}{3}$

We have:

$\phantom{=} 8 - x \gt \dfrac{1}{3} \\[1em] \Rightarrow -x \gt \dfrac{1}{3} - 8 \quad \text{[Subtracting 8 from both sides]} \\[1em] \Rightarrow -x \gt \dfrac{1 - 24}{3} \\[1em] \Rightarrow -x \gt \dfrac{-23}{3} \\[1em] \Rightarrow -x \gt -7.66… \\[1em] \Rightarrow x \lt 7.66… \quad \text{[Multiplying -1 on both sides and reversing the sign]}$

Positive integers less than 7.66.. are {1, 2, 3, 4, 5, 6, 7}

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

(iv) 4(x + 5) < 29

We have:

4(x + 5) < 29

⇒ 4x + 20 < 29

⇒ 4x < 29 - 20 $\quad$ [Subtracting 20 from both sides]

⇒ 4x < 9

⇒ x < $\dfrac{9}{4} \quad \text{[Dividing both sides by 4]}$

⇒ x < 2.25

Positive integers less than 2.25 are {1, 2}

∴ Solution set = {1, 2}

(v) 5 > $\dfrac{2}{3}$x

We have:

5 > $\dfrac{2}{3}$x

⇒ 5 x 3 > 2x $\quad$ [Multiplying 3 on both sides]

⇒ 15 > 2x

⇒ $\dfrac{15}{2}$ > x $\quad$ [Dividing both sides by 2]

⇒ 7.5 > x

⇒ x < 7.5

Positive integers less than 7.5 are {1, 2, 3, 4, 5, 6, 7}

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

(vi) 2 - $\dfrac{7x}{29}$ < $\dfrac{5}{3}$

We have:

$\phantom{=} 2 - \dfrac{7x}{29} \lt \dfrac{5}{3} \\[1em] \Rightarrow -\dfrac{7x}{29} \lt \dfrac{5}{3} - 2 \quad \text{[Subtracting 2 from both sides]} \\[1em] \Rightarrow -\dfrac{7x}{29} \lt \dfrac{5 - 6}{3} \\[1em] \Rightarrow -\dfrac{7x}{29} \lt \dfrac{-1}{3} \\[1em] \Rightarrow -x \lt \dfrac{-1}{3} \times \dfrac{29}{7} \quad \text{[Multiplying} \dfrac{29}{7} \text{ on both sides ]} \\[1em] \Rightarrow -x \lt \dfrac{-29}{21} \\[1em] \Rightarrow -x \lt - 1.38… \\[1em] \Rightarrow x \gt 1.38 \quad \text{[Multiplying -1 on both sides and reversing the sign]}$

Positive integers greater than 1.38 are {2, 3, 4, 5, …}

∴ Solution set = {2, 3, 4, 5, …}