# Inequalities

A comparison relationship between two algebraic expressions or quantities is known as an Inequalities.

There are two main types of inequalities:

1. Greater than or greater than equal to: > or ≥
2.  Less than or less than equal to: < or ≤

Example: 3x + 1 > x - 3 or x2 - 3x ≤ 0

As is the case with equations, inequalities are ordered by degree and by the number of unknowns. In the above two examples, the first is a linear inequality with one unknown and the second is a quadratic inequality with one unknown.

### Solving Inequations

On the basis of the laws of inequality, there are following working rules.

1. Rule of Addition and Subtraction

Adding or subtracting a fixed number to each side of an inequality produces an equivalent inequality.

Example: Adding 2 to each side of the inequality x – 2 ≤ 1 is equivalent to x ≤ 3.

2. Rule of multiplication or division by a positive number

All terms on both sides of an inequality can be multiplied or divided by a positive number.

3. Rule of multiplication or division by a negative number

If all terms on both sides of an inequality are multiplied or divided by a negative number, the sign of the inequality will be reversed.

Example: If 6x – 4 < – 8x + 6 then dividing by -2, it becomes –3x + 2 > 4x – 3

### Solution Sets

The solution of a linear inequation in one variable consists of many real numbers but may not all real numbers. As such the set of solutions is always a subset of real numbers. Solving an inequality means describing a set not just finding a number. This set is the solution set of the problem.

The solution of any inequation is always expressed in a solution set.

1. Ranges where the ends are excluded

If the value of x is denoted as (1, 2) it means 1 < x < 2 i.e. x is greater than 1 but smaller than 2.

2. Ranges where the ends are included

[2, 5] means 2 ≤ x ≤ 5

3. Mixed Ranges

(3, 21] means 3 < x ≤ 21

4. If solution set = {1, 2} means x = 1 and 2 only.

Example 1: Solve for x, -2x > 4

-2x > 4

Divide both sides by -2,

x < -2