Practice Test on Linear Inequation has different types of questions. Students can test their skills and knowledge on **linear inequations** problems by solving all the provided questions on this page. The questions are mainly related to inequalities and finding the solution to the given inequation and draw a graph for the obtained solution set. You can easily draw a graph on a numbered line.

1. Write the equality obtained?

(i) On subtracting 1 from each side 3 > 7

(ii) On adding 3 to each side 12 < 5

(iii) On multiplying (-2) to each side 11 < 4

(iv) On multiplying 4 to each side 15 > 2

Solution:

(i) 3 – 1 > 7 – 1

2 > 6

(ii) 12 + 3 < 5 + 3

15 < 8

(iii) 11 x (-2) 4 x (-2)

-22 < -8

22 > 8

(iv) 15 x 4 > 2 x 4

60 > 8

2. Write the word statement for the following?

(i) x ≥ 15

(ii) x < 2

(iii) x ≤ -5

(iv) x > 16

(v) x ≠ 6

Solution:

(i) The variable x is greater than equal to 15. The possible values of x are 15 and more than 15.

(ii) The variable x is less than 2. The possible values of x are less than 2.

(iii) The variable x is less than and equal to -5. The possible values of x are less than -5.

(iv) The variable x is greater than 16. The possible values of x are more than 16.

(v) The variable x is not equal to 6. The possible values of x are all real numbers other than 6.

3. Find the solution set for each of the following inequations. x ∈ N

(i) x + 5 < 12

(ii) x – 6 > 5

(iii) 5x + 10 ≥ 17

(iv) 2x + 3 ≤ 6

Solution:

(i) x + 5 < 12

Subtract 5 from both sides.

x + 5 – 5 < 12 – 5

x < 7

Replacement set = {1, 2, 3, 4, 5 . .}

Solution set S = {1, 2, 3, 4, 5, 6}

(ii) x – 6 > 5

Add 6 to both sides.

x – 6 + 6 > 5 + 6

x > 11

Replacement set = {1, 2, 3, 4, 5 . .}

Solution set S = {12, 13, 14, 15, . . . }

(iii) 5x + 10 ≥ 17

Subtract 10 from both sides.

5x + 10 – 10 ≥ 17 – 10

5x ≥ 7

Divide 5 by each side.

5x/5 ≥ 7/5

x ≥ 1.4

Replacement set = {1, 2, 3, 4, 5 . .}

Solution set S = {2, 3, 4, 5 . . .}

(iv) 2x + 3 ≤ 6

Subtract 3 from both sides of the inequation

2x + 3 – 3 ≤ 6 – 3

2x ≤ 3

Both sides of the inequation divide by 2.

2x/2 ≤ 3/2

x ≤ 1.5

Replacement set = {1, 2, 3, 4, 5 . .}

Solution set S = {1, 1.5}

4. Find the solution set for each of the following inequations and represent it on the number line.

(i) 3 < x < 10, x ∈ N

(ii) 3x + 2 ≥ 6, x ∈ N

(iii) 3x/2 < 5, x ∈ N

(iv) -4 < 2x/3 + 1 < – 2, x ∈ N

Solution:

(i) 3 < x < 10, x ∈ N

The two cases are 3 < x and x < 10

It can also represent as x > 3 and x < 10

Replacement set = {1, 2, 3, 4, 5 . .}

The solution set for x > 3 is 4, 5, 6, 7 . . . i.e P = {4, 5, 6, 7 . . .}

And the solution set for x < 10 is 1, 2, 3, 4, 5, 6, 7, 8, 9 i.e Q = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Therefore, solution set of the given inequation = P ∩ Q = {4, 5, 6, 7, 8, 9}

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.

(ii) 3x + 2 ≥ 6, x ∈ N

Subtract 2 from both sides

3x + 2 – 2 ≥ 6 – 2

3x ≥ 4

Divide each side by 3

3x/3 ≥ 4/3

x ≥ 1.33

Replacement set = {1, 2, 3, 4, 5 . .}

Solution set S = {2, 3, 4, 5, . . }

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.

(iii) 3x/2 < 5, x ∈ N

Multiply both sides by 2.

3x/2 x 2 < 5 x 2

3x < 10

divide both sides by 3

3x/3 < 10/3

x < 3.33

Replacement set = {1, 2, 3, 4, 5 . .}

Solution Set S = {1, 2}

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.

(iv) -4 < 2x/3 + 1 < – 2, x ∈ N

The two cases are -4 < 2x/3 + 1 and 2x/3 + 1 < – 2

Case I: -4 < 2x/3 + 1

Subtract 1 from both sides

-4 – 1 < 2x/3 + 1 – 1

-5 < 2x/3

Multiply each side by 3

-5 x 3 < 2x/3 x 3

-15 < 2x

Divide each side by 2

-15/2 < 2x/2

-7.5 < x

x > 7.5

Replacement Set = {1, 2, 3, 4, 5 . .}

Solution Set P = {8, 9, 10, 11 . . . }

Case II: 2x/3 + 1 < – 2

Subtract 1 from both sides

2x/3 + 1 – 1 < – 2 – 1

2x/3 < -3

Multiply 3 to both sides

2x/3 x 3 < -3 x 3

2x < -9

Divide both sides by 2

2x/2 < -9/2

x < -4.5

4.5 > x

Replacement set = {1, 2, 3, 4, 5 . .}

Solution set Q = {1, 2, 3}

Therefore, required solution set S = P ∩ Q

S = Null

5. Find the solution set for each of the following and represent the solution set graphically?

(i) x – 6 < 4, x ∈ W

(ii) 6x + 2 ≤ 20, x ∈ W

(iii) 7x + 3 < 5x + 9, x ∈ W

(iv) 3x – 7 > 5x – 1, x ∈ I

Solution:

(i) x – 6 < 4, x ∈ W

Add 6 to both sides

x – 6 + 6 < 4 + 6

x < 10

Replacement set = {0, 1, 2, 3, 4, 5, 6, …}

Therefore, solution set S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.

(ii) 6x + 2 ≤ 20, x ∈ W

Subtract 2 from both sides

6x + 2 – 2 ≤ 20 – 2

6x ≤ 18

Divide each side by 6

6x/6 ≤ 18/6

x ≤ 3

Replacement set = {0, 1, 2, 3, 4, 5, 6, …}

Therefore, solution set S = {0, 1, 2, 3}

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.

(iii) 7x + 3 < 5x + 9, x ∈ W

Move variables to one side and constants to other side of inequation

7x – 5x < 9 – 3

2x < 6

Divide each side by 2

2x/2 < 6/2

x < 3

Replacement set = {0, 1, 2, 3, 4, 5, 6, …}

Therefore, solution set S = {0, 1, 2}

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.

(iv) 3x – 7 > 5x – 1, x ∈ I

Move variables to one side and constants to another side of inequation

-7 + 1 > 5x – 3x

-6 > 2x

divide 2 by each side

-6/2 > 2x/2

-3 > x

Replacement set ={ . . . -4, -3, -2, -1, 0, 1, 2, 3, . . .}

Solution set = { -2, -1, 0, 1, 2, . . . }

Let us represent the solution set graphically.

The solution set is marked on the number line by dots.