On this page, we will learn about what is linear inequality and **linear inequations** and the steps to solve the linear inequalities problems. You will also get the properties of inequation or inequalities. Check out the representation of the solution set on the real line in the following sections. We have provided Solved Problems along with a detailed explanation so that you can better understand the concept.

## What are Linear Inequation and Linear Inequalities?

Linear Inequation is a statement indicating the value of one quantity or algebraic expressions that are not exactly equal to one another.

Inequalities are nothing but the symbols enclosed between two or more algebraic expressions or quantities. The open sentence which involves <, ≤, >, ≥, and ≠ symbols are called the inequalities.

Some of the examples of Linear Inequation are listed below.

- x < 6
- y ≥ 25
- x + 3 > 40
- p ≠ 10

### Linear Inequation

An inequation that contains one variable and that variable highest power is one then is known as the linear inequation in that variable. To make a linear equation into inequation, you have to replace the equal to symbol with the inequality sign. The statements of any of the forms ax + b < 0, ax + b > 0, ax + b≥ 0, ax + b ≤ 0 are the linear inequations invariable x, where a, b are real numbers and a is not equal to zero.

some of the examples of the linear inequation with variable y are included below:

- 3y + 6 ≥ 0
- 9 – y < 0
- 2y > 0
- 25 + 5y ≤ 0

## Questions on Replacement Set or Domain of a Variable

**Example 1.**

Find the replacement set for the inequation x ≤ 9. The replacement set is a set of whole numbers?

**Solution:**

We know that whole numbers W = {0, 1, 2, 3 . . . }.

Replace x with some values of W. Some values of x from W satisfy the inequation and some don’t. Here, the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 satisfy the given inequation x ≤ 9 while the other values don’t.

Thus, the set of all those values of variables that satisfy the given inequation is called the solution set of the given inequation.

Therefore, the solution set for the inequation x ≤ 9 is S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} or S = {x : x ∈ W, x ≤ 9}

**Example 2.**

Find the replacement set for the inequation x > 2. Let the replacement set be the set of natural numbers?

**Solution:**

We know that natural numbers N = {1, 2, 3, 4, 5, 6}

Replace x with some values of N. Some values of x from N satisfy the inequation and some don’t. Here, the values 3, 4, 5, . . . satisfy the given inequation x > 2 while the other values don’t.

Thus, the set of all those values of variables that satisfy the given inequation is called the solution set of the given inequation.

Therefore, the solution set for the inequation x > 2 is S = {3 4, 5, 6, . . . }

**Example 3.**

Find the replacement set and the solution set for the inequation x ≥ -2 when the replacement set is an integer?

**Solution:**

Replacement set I = {. . . -3, -2, -1, 0, 1, 2, 3, . . . }

Solution set S = {-2, -1, 0, 1, 2, . . . } or S = { x : x ∈ I, x ≥ – 2}

**Example 4.**

Find the solution set for the following linear inequations.

(i) x < 5 where replacement set is {-4, -3, -2, -1, 0, 1, 2, 3, 4}

(ii) x ≥ 7 where replacement set is { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(iii) x ≠ 3 where replacement set is { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

**Solution:**

(i) Solution Set S = {-4, -3, -2, -1, 0, 1, 2, 3, 4} S = {x : x ∈ I, -4 < x ≤ 4}

(ii) Solution set S = {7, 8, 9, 10} or S = {x : x ∈ N, 7 < x < 10}

(iii) Solution set S = {0, 1, 2, 4, 5, 6, 7, 8, 9, 10} or S = {x : x ∈ N, x ≠ 3}

### Frequently Asked Questions on Linear Inequality

1. What is the difference between a linear equation and a linear inequality?

The linear equation is an equation that has one or two variables and those exponents are one. Linear inequation also has one variable whose exponent is one. Between two algebraic expressions, the = symbol is enclosed in a linear equation, linear inequality signs are enclosed in a linear inequation. The graph of inequalities is a dashed line but the equation is a solid line in any situation.

2. What is linear inequality?

Linear inequality contains any symbols of inequality. It represents the data that is not equal in graph form. It involves a linear function.