Linear Inequality and Linear Inequations

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.

What are inequalities?

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.

What are inequation?

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.

Linear Inequations:

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.

Domain of the variable or the Replacement set:

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