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

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

**Properties of Inequation or Inequalities**

**Representation of the Solution Set of an Inequation**

**Practice Test on Linear Inequation**

**● **Inequations - Worksheets

**Worksheet on Linear Inequations**

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