# Properties of Inequation or Inequalities

Here we will discuss about properties of inequation or inequalities.

1. The inequation remains unchanged if the same number is added to both sides of inequation.

For example:

(i) x - 2 > 1

⇒ x - 2 + 2 > 1 + 2   (by adding 2 to both sides)

⇒ x > 3

(ii) x < 5

⇒ x + 1 < 5 + 1   (by adding 1 to both sides)

⇒ x + 1 < 6

(iii) x - 3 > 2

⇒ x - 3 + 3 > 2 + 3   (by adding 3 to both sides)

⇒ x > 5

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

For example:

(i) x + 3 ≤ 7

⇒ x + 3 - 3 ≤ 7 - 3   (by subtracting 3 from both sides)

⇒ x ≤ 4

(ii) x ≥ 4

⇒ x - 3 ≥ 4 - 3   (by subtracting 3 from both sides)

⇒ x - 3 ≥ 1

(iii) x + 5 ≤ 9

⇒ x + 5 - 5 ≤ 9 - 5   (by subtracting 5 from both sides)

⇒ x ≤ 4

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

For example:

(i) x/3 < 4

⇒ x/3 × 3 < 4 × 3  (Multiplying 3 to both sides.)

⇒ x < 12

(ii) x/5 < 7

⇒ x/5 × 5 < 7 × 5   (Multiplying 5 to both sides.)

⇒ x < 35

4. The inequation changes if the same negative number is multiplied to both sides of the inequation. It reverses.

For example:

(i) x/5 > 9

⇒ x/5 × (-5) < 9 × (-5)

⇒ -x < -45

⇒ x > 45

(ii) -x > 5

⇒ -x × (-1) < 5 × (-1)

⇒ x < -5

(iii) x/(-2) > 5

⇒ x/(-2) × (-2) < 5 × (-2)

⇒ x < -10

5. The inequation remains unchanged if the same positive number divides both sides of the inequation.

For example:

(i) 2x > 8

⇒ 2x/2 > 8/2   (Dividing both sides by 2)

⇒ x > 4

(ii) 5x > 8

⇒ 5x/5 > 8/5   (Dividing both sides by 5)

⇒ x > 8/5

6. The inequation changes if the same negative number divides both sides. It reverses.

For example:

(i) -3x > 12

⇒ -3x/-3 < 12/-3   (Dividing both sides by -3)

⇒ x < -4

(ii) -5x ≤ -10

⇒ -5x/-5 ≥ -10/-5   (Dividing both sides by -5)

⇒ x ≥ 2

(iii) -4x > 20

⇒ (-4x)/(-4) < 20/(-4)   (Dividing both sides by -4)

⇒ x < -5

### More examples on properties of inequation or inequalities:

Write the inequality obtained for each of the following statements.

(i) On adding 9 to both sides of 21 > 10.

(ii) On multiplying each side of 4 < 12 by -3.

Solution:

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

21 + 9 > 10 + 9

⇒ 30 > 19

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

Therefore, 4 < 12, then 4 × -3 > 12 × -3

⇒ -12 > -36

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

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