# Properties of Inequality

In this chapter we will learn properties of inequality with examples. These properties are important as they help to solve variety of algebra problems.

But let us first review the basics of inequality.

## What is Inequality?

Inequality is used when the given numbers or variables are not equal to each other.

There are 5 symbols used in inequality.

> ⟹ ” greater than “

< ⟹ ” less than “

\mathtt{\geq } ⟹ ” greater than equal to “

\mathtt{\leq } ⟹ ” less than equal to ”

\mathtt{\neq } ⟹ ” not equal to ”

With the help of these 5 symbols, we can do comparison of numbers or variables.

Some examples of inequality are;

(i) 4x + 2 > 3y

Left side of expression ⟹ 4x + 2
Right side of expression ⟹ 3y

It states that left side of expression is greater than right side.

(ii) 21 – 6x < 3

Left side ⟹ 21 – 6x
Right side ⟹ 3

Left side of expression is smaller than right side.

I hope you understood the basic concept of inequality. Let’s move to understand properties of inequality.

## Important properties of inequality

(01) Transitive Property

If three numbers a, b & c are given with following information;

⟹ a > b
⟹ b > c

From the above information, we can automatically infer that a > c.

Similarly for numbers a, b & c it’s given that;

⟹ a < b
⟹ b < c

Then, we can automatically infer that a < c.

(02) Anti Symmetry Property

The values of inequality cannot be interchanged.

Hence, two numbers cannot be greater than and less than each other at the same time.

For example;

If a > b;
Here we cannot interchange the given values.
Hence, writing b > a will be incorrect.

Similarly, if a < b;
then interchanging numbers to write b < a would be incorrect.

(03) Anti reflexive property

A number cannot be greater or less than itself.

hence,
\mathtt{a\ \ \ \cancel{ >} \ \ \ a}\\\ \\ \mathtt{a\ \ \cancel{< } \ \ \ \ a}

(04) Addition property of inequality

In a given inequality we can add same number on both sides without affecting the character of the expression.

a > b

a + c > b + c

Example of Addition property

We know that;
6 > 2

Add number 3 on both sides.

6 + 3 > 2 + 3

9 > 5

We know that 9 is greater than 5.
Hence, the expression is still valid.

(05) Subtraction property of Inequality

In a given inequality we can subtract same number on both sides without affecting the character of given expression.

Hence if;

a > b

Subtracting c on both sides.

a – c > b – c

Example of Subtraction Property.

Consider the below inequality.
10 < 14

Subtracting 6 on both sides.
10 – 6 < 14 – 6

4 < 8

We know that number 4 is less than 8.
Hence, the inequality is still valid.

(06) Multiplication property of Inequality

The property has two parts;
(a) Multiplication of positive number
(b) Multiplication of negative number

(a) Multiplication of positive number

We can multiply both sides of inequality without affecting the character of the expression.

if a > b

Multiply number c on both sides.

ac > bc

Hence, the inequality remains the same.

(b) Multiplication of negative number

Multiplication of same negative number in inequality reverse the sign of given inequality.

If a > b.

Multiply – c on both sides.

then -ac < -bc.

Note that after multiplication ” > ” sign is changed to ” < ” sign.

(07) Division Property of Inequality.

The division property is same as the multiplication property.

It is divided into two part;

(a) Division by positive number
(b) Division by negative number

(a) Division by positive number

Dividing inequality with same number on both sides will not affect the character of the inequality.

if a > b

Dividing both sides by c.

\mathtt{\frac{a}{c} \ >\ \frac{b}{c}}

(b) Division of negative number

Dividing inequality with negative number will reverse the inequality sign.

If a < b

Divide – c on both sides

\mathtt{\frac{-\ a}{c} \ >\ \frac{-\ b}{c}}

Note that sign of inequality changed from ” < ” to ” > “

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