ICSE Class 8 Maths Chapter 21 Linear Inequations Number Line

Chapter 21: Linear Inequations - Number Line

21.1 Introduction

Teacher's Note

Understanding inequations helps in real-world situations like determining safe temperature ranges for food storage or acceptable speed limits on highways.

21.2 Replacement Set and Solution Set

For any linear inequation in x, the set from which the value(s) of variable x is chosen, is called the replacement set or the universal set.

The set of elements of the replacement set (universal set), which satisfy the given inequation, is called the solution set or the truth set.

e.g. Consider the inequation (statement) x > 6;

(i) if replacement set = {2, 4, 6, 8, 10} then, the solution set = {8, 10}

(ii) if replacement set = {1, 3, 5, 7, 9, 11} then, the solution set = {7, 9, 11}

Test Yourself

1. (i) If x ∈ N (Natural numbers) and x < 5; then x = ........., ........., ........., or .........

(ii) If x ∈ W (Whole numbers) and x < 5; then x = ........., ........., ........., ........., or .........

(iii) If x ∈ Z (integers) and - 2 ≤ x < 3; then x = ........., ........., ........., ........., or .........

2. If the replacement set = {-4, -3, -2, -1, 0, 1, 2, 3}, write the solution set for each of the following:

(i) {x : x > 1} = ..................................

(ii) {x : x < 1} = ..................................

(iii) {x : -3 < x ≤ 2} = ..................................

(iv) {x : - 2 ≤ x < 2} = ..................................

Teacher's Note

Replacement and solution sets are used in database filtering and search algorithms, helping computers find relevant results from large datasets.

21.3 Properties

1. Adding the same number to each side of an inequation, does not change the sign of inequality.

i.e. if a > b, then a + c > b + c

and, if a < b, then a + c < b + c.

2. Subtracting the same number from each side of an inequation, does not change the sign of inequality.

i.e. if a > b, then a - c > b - c

and, if a < b, then a - c < b - c.

3. Multiplying each side of an inequation by a positive number, does not change the sign of inequality.

i.e. if a > b and c is positive (i.e. c > 0) then, a.c > b.c

also, if a < b and c > 0; then a.c < b.c.

4. Multiplying each side of an inequation by a negative number, reverses the sign of inequality.

i.e. if a > b and c is negative (i.e. c < 0), then a.c < b.c;

also, if a < b and c < 0; then a.c > b.c.

5. Dividing each side of an inequation by a positive number, does not change the sign of inequality.

i.e. if a > b and c > 0, then \(\frac{a}{c} > \frac{b}{c}\)

also, if a < b and c > 0, then \(\frac{a}{c} < \frac{b}{c}\).

6. Dividing each side of an inequation by a negative number, reverses the sign of inequality.

i.e. if a > b and c < 0, then \(\frac{a}{c} < \frac{b}{c}\)

also, if a < b and c < 0, then \(\frac{a}{c} > \frac{b}{c}\).

Test Yourself

3. 16 > 15 = 16 + 8 > ................... = ............... > ...............

4. 8 < 10 = 8 - 4 ........................ = ...............

5. 3 < 4 = - 5 × 3 ....... - 5 × 4 = ............... > ...............

6. 6 > - 5 = 6 × - 4 ........ - 5 × - 4 = ...............

7. 20 > - 8 = \(\frac{20}{4}\) .......................... = ...............

8. 15 < 21 = \(\frac{15}{-3}\) .......................... = ...............

Teacher's Note

These properties of inequalities are fundamental to solving real-world optimization problems, such as maximizing profits or minimizing costs in business scenarios.

Example 1

Find the solution set of the inequation:

(i) 12 + 6x > 0; where x is a negative integer.

(ii) 30 - 4(2x - 1) < 30; where x is a positive integer.

Solution

(i) 12 + 6x > 0 =

6x > - 12

x > - 2 [Dividing by 6]

∴ x is a negative integer ∴ Solution set = {- 1} (Ans.)

(ii) 30 - 4(2x - 1) < 30 = 30 - 8x + 4 < 30

= 34 - 8x < 30

= - 8x < 30 - 34

= - 8x < - 4

= \(\frac{-8x}{-8}\) > \(\frac{-4}{-8}\) [Dividing by -8]

= x > \(\frac{1}{2}\)

∴ x is a positive integer ∴ Solution set = {1, 2, 3, 4, 5, .............} (Ans.)

Teacher's Note

Solving inequations with real-world constraints (like requiring positive integers) mimics practical decision-making in production planning and resource allocation.

Exercise 21 (A)

1. If the replacement set is the set of natural numbers, solve:

(i) x - 5 < 0 (ii) x + 1 ≤ 7

(iii) 3x - 4 > 6 (iv) 4x + 1 ≥ 17

2. If the replacement set = {- 6, - 3, 0, 3, 6, 9}; find the truth set of the following:

(i) 2x - 1 > 9 (ii) 3x + 7 ≤ 1

3. Solve: 7 > 3x - 8; x ∈ N.

4. Solve: - 17 < 9y - 8; y ∈ Z.

5. Solve: 9x - 7 ≤ 28 + 4x; x ∈ W.

6. Solve: \(\frac{2}{3}\)x + 8 < 12; x ∈ W.

7. Solve: - 5 (x + 4) > 30; x ∈ Z.

8. Solve the inequation 8 - 2x ≥ x - 5; x ∈ N.

9. Solve the inequality 18 - 3 (2x - 5) > 12; x ∈ W.

10. Solve: \(\frac{2x + 1}{3}\) + 15 ≤ 17; x ∈ W.

21.4 Number Line

A number line is a graph (straight line) on which real numbers are marked as shown below:

The solution of every inequation can be represented on a number line.

For example

Thick dots on the number line represent the solution.

The dark arrow on the left side shows that the solution set continues towards left side.

Important

1. For x ≤ 3 where x is a real number; the number line will be as shown below:

The dark circle around 3, shows 3 is included in the solution and the dark line with dark arrow on the left of number 3 shows that every number less than 3 is also included in the solution.

2. For x < 3 where x ∈ R; the number line will be as shown below:

The hollow circle around 3, shows 3 is not included in the solution and the dark line with dark arrow on the left of number 3 shows that every number less than 3 is included in the solution.

Similarly consider the following number lines:

1. [x ≥ 2 and x ∈ R] 2. [x > 2 and x ∈ R]

3. [-1 < x ≤ 3 and x ∈ R] 4. [- 2 ≤ x < 2 and x ∈ R]

5. [- 3 ≤ x < 1 and x ∈ R]

Teacher's Note

Number lines provide visual representations used in engineering and navigation systems to show ranges of acceptable values or safety zones.

Example 2

Graph the solution set on a number line if - 2x + 14 < 6; where x is a real number.

Solution

- 2x + 14 < 6 = - 2x < 6 - 14

= - 2x < - 8

= \(\frac{-2x}{-2}\) > \(\frac{-8}{-2}\) [Division by a negative number, reverses the sign of inequality]

= x > 4

∴ The required graph is: [Number line with hollow circle at 4 and arrow extending right] (Ans.)

Exercise 21 (B)

Solve and graph the solution set on a number line:

1. x - 5 < - 2; x ∈ N

2. 3x - 1 > 5; x ∈ W

3. - 3x + 12 < - 15; x ∈ R

4. 7 ≥ 3x - 8; x ∈ W

5. 8x - 8 ≤ - 24; x ∈ Z

6. 8x - 9 ≥ 35 - 3x; x ∈ N

7. 5x + 4 > 8x - 11; x ∈ Z

8. \(\frac{2x}{5}\) + 1 < - 3; x ∈ R

9. \(\frac{x}{2}\) > - 1 + \(\frac{3x}{4}\); x ∈ N

10. \(\frac{2}{3}\)x + 5 ≤ \(\frac{1}{2}\)x + 6; x ∈ W

11. Solve the inequation 5(x - 2) > 4(x + 3) - 24 and represent its solution on a number line. Given the replacement set is {-4, -3, -2, -1, 0, 1, 2, 3, 4}.

12. Solve \(\frac{2}{3}\)(x - 1) + 4 < 10 and represent its solution on a number line. Given replacement set is {- 8, - 6, - 4, 3, 6, 8, 12}.

13. For each inequation, given below, represent the solution on a number line:

(i) \(\frac{5}{2}\) - 2x ≥ \(\frac{1}{2}\), x ∈ W

(ii) 3(2x - 1) ≥ 2(2x + 3), x ∈ Z

(iii) 2(4 - 3x) ≤ 4(x - 5), x ∈ W

(iv) 4(3x + 1) > 2(4x - 1), x is a negative integer

(v) \(\frac{4 - x}{2}\) < 3, x ∈ R

(vi) -2(x + 8) ≤ 8, x ∈ R

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