In this topic we will learn to solve linear inequality and linear inequations, find the solution, and represent the solution set on the real line.
The open sentence which involves >, ≥, <, ≤ sign are called an inequality. Inequalities can be posed as a question much like equations and solved by similar techniques step-by-step.
A statement indicating that value of one quantity or algebraic expression which is not equal to another are called an inequation.
For example;
(i) x < 5
(ii) x > 4
(iii) 5x ≥ 7
(iv) 3x - 2 ≤ 4
Thus, each of the above statements is an inequation.
An inequation which involves only one variable whose highest power one is known as a linear inequation in that variable.
Linear inequation looks exactly like a linear equation with inequality sign replacing the equality sign.
The statements of any of the forms ax + b > 0, ax + b ≥ 0, ax + b < 0, ax + b ≤ 0 are linear inequations in variable x, where a, b are real numbers and a ≠ 0.
For example;
(i) 2x + 1 > 0,
(ii) 5x ≤ 0,
(iii) 5 - 4x < 0,
(iv) 9x ≥ 0
Thus, each of the above statement is linear inequation in variable x.
For a given inequation, the set from which the values of the variable are replaced is called
domain of the variable or the replacement set.
For example;
1. Consider an inequation x < 4. Let the replacement be the set of whole numbers (W).
Solution:
We know that W = {0, 1, 2, 3, ...}. We replace x by some values of W. Some values of x from W satisfy the inequation and some don’t. Here, the values 0, 1, 2, 3 satisfy the given inequation x < 4 while the other values don’t.
Thus, the set of all those values of variables which satisfy the given inequation is called the solution set of the given inequation.
Note:
Every solution set is a subset of replacement set.
Therefore, the solution set for the inequation x < 4 is S = {0, 1, 2, 3} or S = {x : x ∈ w, x < 4}
2. Consider an inequation x < 5. Let the replacement set be the set of natural numbers (N). Solution:
We know that N = {1, 2, 3, 4, 5, 6, ...}. We replace x by some values of N which satisfy the given inequation. These values are 1, 2, 3, 4.
Thus, a solution set of all those values of variables which satisfy the given inequation is called the solution set of the given inequation.
Note:
Every solution set is a subset of replacement set.
Therefore, the solution set for the inequation x < 5, x ∈ N is S = {1, 2, 3,} or S {x : x ∈ N, x < 5}.
3. Find the replacement set and the solution set for the inequation x ≥ -2 when replacement set is an integer.
Solution:
Replacement set = {... -3, -2, -1, 0, 1, 2, 3, ...}
Solution set = {-2, -1, 0, 1, 2, ...} or S = {x : x ∈ I, x ≥ -2}
4. Find the solution set for the following linear inequations.
(i) x > -3 where replacement set is S = {-4, -3, -2, -1, 0, 1, 2, 3, 4}
(ii) x ≤ -2 where replacement set {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4}
Solution:
(i) Solution set S = {-2, -1, 0, 1, 2, 3, 4} or S = (x : x ∈ I, -3 < x ≤ 4}
(ii) Solution set S = {-2, -3, -4, -5} or S = {x : x ∈ I,- 5 < x ≤ - 2
● Inequations
Properties of Inequation or Inequalities
Representation of the Solution Set of an Inequation
Practice Test on Linear Inequation
● Inequations - Worksheets
Worksheet on Linear Inequations
7th Grade Math Problems
8th Grade Math Practice
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