Page 158 - ICSE Math 7

14 Linear Inequations

Key Concepts

• Inequation • Method of Solving Inequations
• Replacement and Solution Sets • Graphical Representation of Solution Set

Inequation

An inequation is a statement which states that the two algebraic expressions are not equal. The symbol
‘>’ stands for ‘greater than’, ‘<’ stands for ‘less than’, ‘≥’ stands for ‘greater than or equal to’ and
‘≤’ stands for ‘less than or equal to’. An inequation in which the highest power of the variable is 1 is
known as a linear inequation. For example: x < 5 and 3x – 4 > 9 – x are inequations.

Points to remember
• An inequation remains unchanged while solving if:
(a) the same number is added to both the sides
(b) the same number is subtracted from both the sides
(c) both the sides are multiplied or divided by the same positive number
• If both the sides of an inequation are multiplied or divided by the same negative number, then
the sign of inequality is reversed, i.e., ‘>’ becomes ‘<’, ‘<’ becomes ‘>’, ‘≥’ becomes ‘≤’ and ‘≤’
becomes ‘≥’.
• A linear inequation may have many or even infinite solutions.

Replacement Set
The set from which the values of the variable have to be selected to make an inequation true is known
as the replacement set or domain of the variable. Example:
(a) If x < 4 and x ∈ W, then the values of x have to be chosen from the set of whole numbers (W).
So, W is the replacement set for this inequation.
(b) If z > –3 and z ∈ {–5, –4, –2, 1, 13}, then for this inequation the values of z have to be chosen
from the set {–5, –4, –2, 1, 13}. So, {–5, –4, –2, 1, 13} is the replacement set for this inequation.
Solution Set
The values of the variable from the replacement set which satisfy Maths Info
the inequation form the solution set or truth set of the inequation. Solution set is always a subset
Example: of the replacement set.
(a) If x < 4 and x ∈ W, then
Replacement set = W = {0, 1, 2, 3, 4, …}
But, only the values 0, 1, 2 and 3 from the replacement set satisfy the inequation x < 4.
\ Solution set = {0, 1, 2, 3}

(b) If z > – 3 and z ∈{–5, – 4, –2, 1, 13}, then
Replacement set = {–5, –4, –2, 1, 13}
But, only the values –2, 1 and 13 from the replacement set satisfy the inequation z > –3.
\ Solution set = {–2, 1, 13}

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