Properties of Inequation or Inequalities page gives detailed information about six properties. Each Property is explained step by step by considering a few examples. Learn and understand the properties easily with the help of solved examples provided below. You can seek Complete Information about **Linear Inequations** and their definitions all on our page.

## Properties of Inequation or Inequalities

The six different properties of linear inequalities or linear inequation are mentioned here. These properties describe how arithmetic operations show the effect on the linear inequations.

**Property I**

The inequation remains unchanged if the same quantity is added to both sides of it.

Example:

x – 3 < 4

Add 3 to the both sides

= x – 3 + 3 < 4 + 3

= x < 7

**Property II:**

The inequation remains unchanged if the same quantity is subtracted from the both sides of the inequation.

Example:

x + 3 < 4

Subtract 3 from both sides

x + 3 – 3 < 4 – 3

x < 1

**Property III:**

The inequation remains unchanged if the same positive number is multiplied to both sides of it.

Example:

x/3 > 5

multiply 3 to both sides

= x/3 x 3 > 5 x 3

= x > 15

**Property IV:**

The inequation changes if the same negative number is multiplied to both sides of it. Actually, it reverses.

Example:

x/5 < 6

multiply -5 to the both sides

= x/5 x (-5) < 6 x (-5)

= -x < -30

= x > 30

**Property V:**

The inequation remains unchanged if the same positive number is divided by both sides of the inequation.

Example:

5x > 20

dividing both sides by 5

= 5x/5 > 20/5

= x > 4

**Property VI: **

The inequation changes when the same negative number is divided by both sides. It reverses.

Example:

-3x < 12

Dividing both sides by -3

= -3x/-3 > 12/-3

= x > -4

### Solved Examples on Properties of Linear Inequalities

**Example 1.**

Write the inequality obtained for each of the following statements.

(i) On subtracting 9 from both sides of 21 > 10.

(ii) On multiplying each side of 8 < 12 by -2.

**Solution:**

(i) We know that subtracting the same number from both sides of inequality does not change the inequality.

21 – 9 > 10 – 9

= 12 > 1

(ii) We know that multiplying each side of an inequality by the same negative number reverses that inequality.

= 8 x -2 < 12 x -2

= -16 < -24

= -16 > -24

**Example 2.**

Find the inequality obtained for the following statements.

(i) On adding 2 to both sides of 8 < 56

(ii) On dividing each side of 8 < 56 by -16.

**Solution:**

(i) We know that adding the same number to both sides of inequality does not change the inequality.

8 + 2 < 56 + 2

= 10 < 58

(ii) We know that dividing each side of an inequality by the same negative number reverses that.

8/-16 < 56/-16

= -1/2 > -7/2

**Example 3.**

Write the inequality obtained for the following statement.

On multiplying each side of 18 > 8 by 5.

**Solution:**

We know that multiplying each side of an inequality by the same positive number does not change the inequality.

18 x 5 > 8 x 5

= 90 > 40