**Solving Linear Inequalities One Variable Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice solving linear inequalities in one variable.

**Problem 1 : **

Solve for x :

5x - 3 < 3x + 1

(i) when x is a real number

(ii) when x is integer number

(iii) when x is a natural number

**Problem 2 : **

Solve for x :

3x + 17 ≤ 2(1 - x)

**Problem 3 :**

Solve for x :

2(2x + 3) - 10 ≤ 6(x - 2)

**Problem 4 : **

Solve for x :

3x - 7 > x + 1

**Problem 5 :**

Solve for x :

- (x - 3) + 4 < 5 - 2x

**Problem 1 :**

Solve 5x - 3 < 3x + 1 when

(i) when x is a real number

(ii) when x is an integer

(iii) when x is a natural number

**Solution :**

**(i) When x is a real number :**

5x - 3 < 3x + 1

Subtract 3x from each side.

2x - 3 < 1

Add 3 to each side.

2x < 4

Divide each side by 2

x < 2

Because x is real number, the solution set is

(-∞, 2)

**(ii) When x is an integer :**

We have already solved for x in the given inequality.

That is

x < 2

Because x is an integer, the solution set is

{...............,-4, -3, -2, - 1, 0, 1, 2, 3,...............}

**(iii) When x is a natural number : **

x < 2

Because x is a integer, the solution set is

{ 1 }

**Problem 2 :**

Solve for x :

3x + 17 ≤ 2(1 - x)

**Solution :**

3x + 17 ≤ 2(1 - x)

3x + 17 ≤ 2 - 2x

Add 2x to each side.

5x + 17 ≤ 2

Subtract 17 from each side.

5x ≤ - 15

Divide each side by 5.

x ≤ - 3

So, the solution set is

(-∞, -3]

**Problem 3 :**

Solve for x :

2(2x + 3) - 10 ≤ 6(x - 2)

**Solution :**

2(2x + 3) - 10 ≤ 6(x - 2)

4x + 6 - 10 ≤ 6x - 12

4x - 4 ≤ 6x - 12

Subtract 6x from each side.

-2x - 4 ≤ - 12

Add 4 to each side.

-2x ≤ - 8

Divide each side by (-2).

x ≥ 4

So, the solution set is

[4, ∞)

**Problem 4 :**

Solve for x :

3x - 7 > x + 1

**Solution :**

3x - 7 > x + 1

Subtract x from each side.

2x - 7 > 1

Add 7 to each side.

2x > 8

Divide each side by 2.

x > 4

So, the solution set is

(4, ∞)

**Problem 5 :**

Solve for x :

- (x - 3) + 4 < 5 - 2x

**Solution :**

- ( x - 3) + 4 < 5 - 2x

-x + 3 + 4 < 5 - 2x

-x + 7 < 5 -2x

Add 2x to each side.

x + 7 < 5

Subtract 7 from each side.

x < - 2

So, the solution set is (-∞, -2).

After having gone through the stuff given above, we hope that the students would have understood, how to solve linear inequalities in one variable.

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