# 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

Inequations

What are Linear Inequality?

What are Linear Inequations?

Properties of Inequation or Inequalities

Representation of the Solution Set of an Inequation

Inequations - Worksheets

Worksheet on Linear Inequations