Inequalities - Exercise 15(A)

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

Answer

(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:

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

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:

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

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

∴ Solution set = {1, 2, 3, 4, 5}

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

Answer

(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

Multiply -1 on both sides and reverse the sign

⇒ x > 5

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:

$\phantom{=} \dfrac{4}{7}-\dfrac{x}{4} \gt - 2 \\[1em] \text{Subtract } \dfrac{4}{7} \text{ from both sides} \\[1em] \Rightarrow -\dfrac{x}{4} \gt -2 - \dfrac{4}{7} \\[1em] \Rightarrow -\dfrac{x}{4} \gt \dfrac{-14 - 4}{7} \\[1em] \Rightarrow -\dfrac{x}{4} \gt \dfrac{-18}{7} \\[1em] \text{Multiply 4 on both sides} \\[1em] \Rightarrow -x \gt \dfrac{-18}{7} \times 4 \\[1em] \Rightarrow -x \gt \dfrac{-72}{7} \\[1em] \Rightarrow -x \gt -10.28... \\[1em] \text{[Multiply -1 on both sides and reverse the sign]} \\[1em] \Rightarrow x \lt 10.28...$

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:

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

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

∴ Solution set = {3, 4, 5, ...}

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 - $\dfrac{7x}{29}$ < $\dfrac{5}{3}$

Answer

(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:

$\phantom{=} 8 - x \gt \dfrac{1}{3} \\[1em] \text{Subtract 8 from both sides} \\[1em] \Rightarrow -x \gt \dfrac{1}{3} - 8 \\[1em] \Rightarrow -x \gt \dfrac{1 - 24}{3} \\[1em] \Rightarrow -x \gt \dfrac{-23}{3} \\[1em] \Rightarrow -x \gt -7.66... \\[1em] \text{Multiply -1 on both sides and reverse the sign} \\[1em] \Rightarrow x \lt 7.66...$

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:

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

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

∴ Solution set = {2, 3, 4, 5, ...}

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

Answer

(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

Subtract x and 5 from both sides

⇒ 6x - x > - 10 - 5

⇒ 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

Multiply -1 on both sides and reverse the sign

⇒ x > -2.25

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

∴ Solution set = {-2, -1}

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

We have:

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

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

∴ Solution set = {..., -6, -5}

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

Answer

(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

Subtract 19 and add 2x on both sides

⇒ 4 - 19 < 3x + 2x

⇒ -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

Subtract 19 and add 5x on both sides

⇒ 3x + 5x < 42 - 19

⇒ 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:

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

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:

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

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

∴ Solution set A = {4, 5, 6, ...}

Case 2:

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

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

∴ Solution set B = {..., 3, 4, 5}

Final solution set = A ∩ B = {4, 5}