If x ∈ {-2, -1, 0, 1, 2, 3, 4, 5}, find the solution set of each of the following inequations :

(i) 2x > 5

(ii) 3x - 8 < 1

(iii) 3 - 12x > -21

(iv) 7 - x > 0

(v) 3 - 4x > -2

(vi) 3x + 4 < 15

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

(viii) $\dfrac{2}{3} + x$ < - $\dfrac{1}{6}$

(ix) $\dfrac{7}{4} - 3x$ < $\dfrac{5}{6}$

(i) 2x > 5

We have :

2x > 5

⇒ x > $\dfrac{5}{2} \quad \text{[Dividing both sides by 2]}$

⇒ x > 2.5

From the set, values greater than 2.5 are {3, 4, 5}.

∴ Solution set = {3, 4, 5}

(ii) 3x - 8 < 1

We have:

3x - 8 < 1

⇒ 3x < 1 + 8 $\quad$ [Adding 8 on both sides]

⇒ 3x < 9

⇒ x < $\dfrac{9}{3} \quad \text{[Dividing both sides by 3]}$

⇒ x < 3

From the set, values less than 3 are {-2, -1, 0, 1, 2}.

∴ Solution set = {-2, -1, 0, 1, 2}

(iii) 3 - 12x > -21

We have:

3 - 12x > -21

⇒ -12x > -21 - 3 $\quad$ [Subtracting 3 from both sides]

⇒ -12x > -24

Dividing by a negative number reverses the sign:

⇒ x < $\dfrac{-24}{-12} \quad \text{[Dividing both sides by -12]}$

⇒ x < 2

From the set, values less than 2 are {-2, -1, 0, 1}.

∴ Solution set = {-2, -1, 0, 1}

(iv) 7 - x > 0

We have:

7 - x > 0

⇒ 7 > x

⇒ x < 7

All values in the set are less than 7.

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

(v) 3 - 4x > -2

We have:

3 - 4x > -2

⇒ -4x > -2 - 3 $\quad$ [Subtracting 3 from both sides]

⇒ -4x > -5

Dividing by a negative number reverses the sign:

⇒ x < $\dfrac{-5}{-4} \quad \text{[Dividing both sides by -4]}$

⇒ x < 1.25

From the set, values less than 1.25 are {-2, -1, 0, 1}.

∴ Solution set = {-2, -1, 0, 1}

(vi) 3x + 4 < 15

We have:

3x + 4 < 15

⇒ 3x < 15 - 4 $\quad$ [Subtracting 4 from both sides]

⇒ 3x < 11

⇒ x < $\dfrac{11}{3} \quad \text{[Dividing both sides by 3]}$

⇒ x < 3.66…

From the set, values less than 3.66… are {-2, -1, 0, 1, 2, 3}.

∴ Solution set = {-2, -1, 0, 1, 2, 3}

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

We have:

$\dfrac{3}{4}x$ > - 1

⇒ 3x > -1 x 4 $\quad$ [Multiplying 4 on both sides]

⇒ 3x > -4

⇒ x > $\dfrac{-4}{3} \quad \text{[Dividing both sides by 3]}$

⇒ x > -1.33…

From the set, values greater than -1.33… are {-1, 0, 1, 2, 3, 4, 5}.

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

(viii) $\dfrac{2}{3} + x$ < - $\dfrac{1}{6}$

We have:

$\phantom{=} \dfrac{2}{3} + x \lt - \dfrac{1}{6} \\[1em] \Rightarrow x \lt - \dfrac{1}{6} - \dfrac{2}{3} \quad \text{[Subtracting} \dfrac{2}{3} \text{ from both sides]} \\[1em] \Rightarrow x \lt \dfrac{-1 - 4}{6} \\[1em] \Rightarrow x \lt \dfrac{-5}{6} \\[1em] \Rightarrow x \lt -0.833…$

From the set, values less than -0.833 are {-2 , -1}.

∴ Solution set = {-2, -1}

(ix) $\dfrac{7}{4} - 3x$ < $\dfrac{5}{6}$

We have:

$\phantom{=} \dfrac{7}{4} - 3x \lt \dfrac{5}{6} \\[1em] \Rightarrow -3x \lt \dfrac{5}{6} - \dfrac{7}{4} \quad \text{[Subtracting } \dfrac{7}{4} \text{ from both sides]} \\[1em] \Rightarrow -3x \lt \dfrac{5}{6} - \dfrac{7}{4} \\[1em] \Rightarrow -3x \lt \dfrac{10 - 21}{12} \\[1em] \Rightarrow -3x \lt \dfrac{-11}{12} \\[1em] \Rightarrow x \gt \dfrac{-11}{12 \times (-3)} \quad \text{[Dividing both sides by -3 and reversing the sign]} \\[1em] \Rightarrow x \gt \dfrac{-11}{-36} \\[1em] \Rightarrow x \gt 0.305…$

From the set, values greater than 0.305… are {1, 2, 3, 4, 5}.

∴ Solution set = {1, 2, 3, 4, 5}