Page 173 - ICSE Math 8

For example, (i) If a + 3 > 12, then a + 3 – 3 > 12 – 3 ⇒ a > 9
(ii) If a – 4 < 3, then a – 4 + 4 < 3 + 4 ⇒ a < 7
(b) The symbol of inequality remains unchanged if both sides are multiplied or divided by the same positive
number. Symbolically, this can be represented as:
b
a
If a > b, then ac > bc or > , where c is a positive number.
c c
x x
For example, (i) If ≥ 3, then × 5 ≥ 3 × 5 ⇒ x ≥ 15
5 5
7y 14
(ii) If 7y < 14, then < ⇒ y < 2
7 7
(c) The symbol of inequality is reversed if both sides are multiplied or divided by the same negative number.
Symbolically, this can be represented as:
a
b
If a > b, then ac < bc or < , where c is a negative number.
c c
For example, (i) If –x > 4, then –2 × –x < 4 × (–2) ⇒ x < –8
2 2

(ii) If –3x > –9, then –3x < –9 ⇒ x < 3
–3 –3
(d) Any term from one side of an inequation can be shifted to the other side by changing its sign. This is
known as rule of transposition.
For example, (i) If 2x = 9 – 5x, then 2x + 5x = 9
(ii) If 3x + 5 = 2y – 9, then 3x = 2y – 9 – 5 ⇒ 3x = 2y – 14

Method of Solving Inequations

To solve an inequation, follow these steps.
Step 1: Simplify both the sides of the inequation.
Step 2: Using the rule of transposition, transpose the terms containing the variable on one side and the
constant terms on the other side.
Step 3: Divide both sides of the inequation by the coefficient of the variable.
Step 4: Form the solution set from the replacement set.

Example 16: Find the solution set of the inequation 20 – 3(2x – 4) > 14; Try This
x ∈ W.
Solution: 20 – 3(2x – 4) > 14 ⇒ 20 – 6x + 12 > 14 Solve the following inequations.

⇒ –6x > 14 – 20 – 12 (Using the rule of transposition) (a) 4 – 3x ≤ –5, x ∈ {2, 4, 6, 8}
13
3
⇒ –6x > –18 (b) 2 < 10x + , x ∈ {–1, 0, 1}
2
⇒ –6x < –18 ⇒ x < 3 (Dividing both sides by –6 and reversing the sign)
–6 –6
As the replacement set is W, x should be a whole number less than 3.
∴ Solution set = {0, 1, 2}

Graphical Representation of Solution
We can represent the solution set graphically by plotting points on the number line.
Example 17: Find the solution set for the inequation 3x + 2 ≤ 8. If the replacement set is {1, 2, 3, 4, 5, 6, 7}
represent the solution on a number line.

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