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

(i) 3x > - 14

(ii) -29 < 9x - 2

(iii) -4(x + 5) < 9

(iv) 5 + 6x > x - 10

(v) 10 - 2(1 + 4x) < 26

(vi) $\dfrac{1}{3}$ > $\dfrac{6}{7}x + 4$

(i) 3x > - 14

We have:

3x > - 14

⇒ x > $\dfrac{-14}{3} \quad \text{[Dividing both sides by 3]}$

⇒ x > -4.66…

Negative integers greater than -4.66.. are {-4, -3, -2, -1}

∴ Solution set = {-4, -3, -2, -1}

(ii) -29 < 9x - 2

We have:

-29 < 9x - 2

⇒ -29 + 2 < 9x $\quad$ [Adding 2 on both sides]

⇒ -27 < 9x

⇒ $\dfrac{-27}{9}$ < x $\quad$ [Dividing both sides by 9]

⇒ -3 < x

⇒ x > -3

Negative integers greater than -3 are {-2, -1}

∴ Solution set = {-2, -1}

(iii) -4(x + 5) < 9

We have:

-4(x + 5) < 9

⇒ -4x - 20 < 9

⇒ -4x < 9 + 20 $\quad$ [Adding 20 on both sides]

⇒ -4x < 29

⇒ -x < $\dfrac{29}{4} \quad \text{[Dividing both sides by 4]}$

⇒ -x < 7.25

⇒ x > -7.25 $\quad$ [Multiplying -1 on both sides and reversing the sign]

Negative integers greater than -7.25 are {-6, -5, -4, -3, -2, -1}

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

(iv) 5 + 6x > x - 10

We have:

5 + 6x > x - 10

⇒ 6x - x > - 10 - 5 $\quad$ [Subtracting x and 5 from both sides]

⇒ 5x > -15

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

⇒ x > -3

Negative integers greater than -3 are {-2, -1}

∴ Solution set = {-2, -1}

(v) 10 - 2(1 + 4x) < 26

We have:

10 - 2(1 + 4x) < 26

⇒ 10 - 2 - 8x < 26

⇒ 8 - 8x < 26

⇒ -8x < 26 - 8 $\quad$ [Subtracting 8 from both sides]

⇒ -8x < 18

⇒ -x < $\dfrac{18}{8} \quad \text{[Dividing both sides by 8]}$

⇒ -x < 2.25

⇒ x > -2.25 $\quad$ [Multiplying -1 on both sides and reversing the sign]

Negative integers greater than -2.25 are {-2, -1}

∴ Solution set = {-2, -1}

(vi) $\dfrac{1}{3}$ > $\dfrac{6}{7}x + 4$

We have:

$\phantom{=} \dfrac{1}{3} \gt \dfrac{6}{7}x + 4 \\[1em] \Rightarrow \dfrac{1}{3} - 4 \gt \dfrac{6}{7}x \quad \text{[Subtracting 4 from both sides]} \\[1em] \Rightarrow \dfrac{1 - 12}{3} \gt \dfrac{6}{7}x \\[1em] \Rightarrow \dfrac{-11}{3} \gt \dfrac{6}{7}x \\[1em] \Rightarrow \dfrac{7}{6} \times \dfrac{-11}{3} \gt x \quad \text{[Multiplying } \dfrac{7}{6} \text{ on both sides]} \\[1em] \Rightarrow \dfrac{-77}{18} \gt x \\[1em] \Rightarrow -4.27… \gt x \\[1em] \Rightarrow x \lt -4.27…$

Negative integers less than -4.27… are {…, -6, -5}

∴ Solution set = {…, -6, -5}