# Problems on Linear Inequation | Solving Linear Inequalities Word Problems with Answers PDF

Problems on Linear Inequation with answers and solving procedures are given here. Make the use of our Problems on Linear Inequation Problems and solving process and practice well. Students who are preparing for the 10th grade Math exam can get these practice questions on linear inequations from here and use them online or offline. The entire Linear Inequation concepts, worksheets, problems, and their solution pdfs available here are free of cost. So, without any delay begin your practice and prepare well for the exam.

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## Examples of Linear Inequalities with Solutions

Question 1. Solve 8x – 16 ≤ 24

Solution:

(i) Given Inequation is 8x – 16 ≤ 24 where x is the variable.
8x – 16 ≤ 24
Now, add 16 on both sides of the above equation.
8x – 16 + 16 ≤ 24 + 16
8x ≤ 40
Now, divide the above equation with 8 into both sides.
8x/8 ≤ 40/8
x ≤ 5.
The value of x is 5 or less than 5.

Therefore, the answer is x ≤ 5.

Question 2. Solve the inequation 4(x – 4) ≥ 6x – 10

Solution:

Given Inequation is 4(x – 4) ≥ 6x – 10 where x is the variable.
4(x – 4) ≥ 6x – 10
4x – 16 ≥ 6x – 10
Move the 4x to the right side of the above equation.
– 16 ≥ 6x – 4x – 10
-16 ≥ 2x – 10
Now, add 10 on both sides of the above equation.
-16 + 10 ≥ 2x – 10 + 10
-6 ≥ 2x
Now, divide the above equation with 2 into both sides.
-6/2 ≥ 2x/2
-3 ≥ x
x ≤ -3
The value of x is -3 or less than -3.

Therefore, the answer is x ≤ -3.

Question 3. Solve the inequation: – 10 ≤ 4x – 14 ≤ 2

Solution:

Given Inequation is – 10 ≤ 4x – 14 ≤ 2 where x is the variable.
– 10 ≤ 4x – 14 …….. (i)
4x – 14 ≤ 2 …………..(ii)
Firstly, solve the equation (i)
– 10 ≤ 4x – 14 …….. (i)
Now, add 14 on both sides of the above equation.
– 10 + 14 ≤ 4x – 14 + 14
4 ≤ 4x
Now, divide the above equation with 4 into both sides.
4/4 ≤ 4x/4
1 ≤ x
x ≥ 1.
The value of x is 1 or more than 1.
Now, solve the equation (ii)
4x – 14 ≤ 2 …………..(ii)
Now, add 14 on both sides of the above equation.
4x – 14 + 14 ≤ 2 + 14
4x ≤ 16
Now, divide the above equation with 4 into both sides.
4x/4 ≤ 16/4
x ≤ 4
The value of x is 4 or less than 4.

Therefore, the answer is 1 ≤ x ≤ 4. Here least value of x is 1 and the greatest value of x is 4.

Question 4. Solve the following inequation 4x – 10 ≤ 10x + 8 < 22, where x ∈ I.

Solution:

Given Inequation is – 10 ≤ 4x – 14 ≤ 2 where x is the variable.
– 10 ≤ 4x – 14 …….. (i)
4x – 14 ≤ 2 …………..(ii)
Firstly, solve the equation (i)
– 10 ≤ 4x – 14 …….. (i)
Now, add 14 on both sides of the above equation.
– 10 + 14 ≤ 4x – 14 + 14
4 ≤ 4x
Now, divide the above equation with 4 into both sides.
4/4 ≤ 4x/4
1 ≤ x
x ≥ 1.
The value of x is 1 or more than 1.
Now, solve the equation (ii)
4x – 14 ≤ 2 …………..(ii)
Now, add 14 on both sides of the above equation.
4x – 14 + 14 ≤ 2 + 14
4x ≤ 16
Now, divide the above equation with 4 into both sides.
4x/4 ≤ 16/4
x ≤ 4
The value of x is 4 or less than 4.

Therefore, the answer is 1 ≤ x ≤ 4. Here least value of x is 1 and the greatest value of x is 4.

Question 5. Solve the following inequation and graph the solution on a number line 4x – 10 ≤ 10x + 8 < 22, where x ∈ I.

Solution:

Given Inequation is 4x – 10 ≤ 10x + 8 < 22, where x ∈ I.
4x – 10 ≤ 10x + 8 …….. (i)
10x + 8 < 22 …………..(ii)
Firstly, solve the equation (i)
4x – 10 ≤ 10x + 8 …….. (i)
Now, move the 10x to the left side of the above equation.
4x -10x – 10 ≤ 8
-6x – 10 ≤ 8
Now, add 10 on both sides of the above equation.
-6x – 10 + 10 ≤ 8 + 10
-6x ≤ 18
Now, divide the above equation with -6 into both sides. The inequality reverses on multiplying both sides by -1.
-6x/-6 ≤ 18/-6
x ≥ -3
The value of x is -3 or more than -3.
Now, solve the equation (ii)
10x + 8 < 22 …………..(ii)
Now, subtract 8 from both sides of the above equation.
10x + 8 – 8 < 22 – 8
10x < 14
Now, divide the above equation with 10 into both sides.
10x/10 < 14/10
x < 7/5
The value of x is less than 7/5.

Therefore, the answer is -3 ≤ x < 7/5. Here least value of x is -3 and the greatest value of x is 7/5. So, the x ∈ {-3, -2, -1, 0, 1}. The graph of the solution on a number line is

Question 6. Given that x ∈ I, solve the inequation 6 ≥ (x – 4) + 2x/3 ≥ 4

Solution:

Given Inequation is 6 ≥ (x – 4) + 2x/3 ≥ 4, where x ∈ I.
6 ≥ (x – 4) + 2x/3 …….. (i)
(x – 4) + 2x/3 ≥ 4 …………..(ii)
Firstly, solve the equation (i)
6 ≥ (x – 4) + 2x/3 …….. (i)
6 ≥ x – 4 + 2x/3
Now, solve the x terms.
6 ≥ 5x/3 – 4
Now, add 4 on both sides of the above equation.
6 + 4 ≥ 5x/3 – 4 + 4
10 ≥ 5x/3
Now, multiply the above equation with 3 on both sides.
30 ≥ 5x
Now, divide the above equation with 5 into both sides.
30/5 ≥ 5x/5
6 ≥ x
x ≤ 6.
The value of x is 6 or less than 6.
Now, solve the equation (ii)
(x – 4) + 2x/3 ≥ 4 …………..(ii)
Now, solve the x terms.
5x/3 – 4 ≥ 4
Now, add 4 on both sides of the above equation.
5x/3 – 4 + 4 ≥ 4 + 4
5x/3 ≥ 8
Now, multiply the above equation with 3 on both sides.
5x ≥ 24
Now, divide the above equation with 5 into both sides.
5x/5 ≥ 24/5
x ≥ 4.8
The value of x is 4.8 or more than 4.8.

Therefore, the answer is 4.8 ≤ x ≤ 6. Here least value of x is 4.8 and the greatest value of x is 6. So, the x ∈ {5, 6}.

Question 7. Given A = {x : 22x – 10 > 14x + 6, x ∈ R}, B = {x : 36x – 18 ≥ 30 + 24x, x ∈ R}. Find the range of the set A ∩ B.

Solution:

Given Inequations are Given A = {x : 22x – 10 > 14x + 6, x ∈ R}, B = {x : 36x – 18 ≥ 30 + 24x, x ∈ R}.
Firstly, solve the equation A = {x : 22x – 10 > 14x + 6, x ∈ R}
22x – 10 > 14x + 6
Now, solve the x terms. Move 14x to the left side of the above equation.
22x – 14x – 10 > 6
8x – 10 > 6
Now, add 10 on both sides of the above equation.
8x – 10 + 10 > 6 + 10
8x > 16
Now, divide the above equation with 8 into both sides.
8x/8 > 16/8
x > 2
The value of x is greater than 2.
Now, solve the equation B = {x : 36x – 18 ≥ 30 + 24x, x ∈ R}
36x – 18 ≥ 30 + 24x
Now, solve the x terms. Move 24x to the left side of the above equation.
36x -24x – 18 ≥ 30
12x – 18 ≥ 30
Now, add 18 on both sides of the above equation.
12x – 18 + 18 ≥ 30 + 18
12x ≥ 48
Now, divide the above equation with 12 into both sides.
12x/12 ≥ 48/12
x ≥ 4
The value of x is 4 or more than 4.

Therefore, the answer is 4 ≤ x < 6. Here least value of x is 4 and the greatest value of x is 6. So, the range of the set A ∩ B = {x : x ≥ 4, x ∈ R}.

Question 8. Solve the given inequation 4y – 6 < 2y + 2 ≤ 8y + 14, y ∈ R

Solution:

Given Inequation is 4y – 6 < 2y + 2 ≤ 8y + 14, y ∈ R.
4y – 6 < 2y + 2 …….. (i)
2y + 2 ≤ 8y + 14 …………..(ii)
Firstly, solve the equation (i)
4y – 6 < 2y + 2 …….. (i)
Now, solve the y terms. Move 2y to the left side of the above equation.
4y -2y – 6 < 2
2y – 6 < 2
Now, add 6 on both sides of the above equation.
2y – 6 + 6 < 2 + 6
2y < 8
Now, divide the above equation with 2 into both sides.
2y/2 < 8/2
y < 4
The value of y is less than 4.
Now, solve the equation (ii)
2y + 2 ≤ 8y + 14…………..(ii)
Now, solve the y terms. Move 2y to the right side of the above equation.
2 ≤ 8y -2y + 14
2 ≤ 6y + 14
Now, subtract 14 from both sides of the above equation.
2 – 14 ≤ 6y + 14 – 14
– 12 ≤ 6y
Now, divide the above equation with 6 into both sides.
– 12/6 ≤ 6y/6
-2 ≤ y
y ≥ -2
The value of y is -2 or more than -2.

Therefore, the answer is -2 ≤ y < 4. Here least value of y is -2 and the greatest value of y is 4. So, the answer is {y : -2 ≤ y < 4 , y ∈ R}.

Question 9. Solve the given inequation and graph it on a number line -6 < -1 – $$\frac { 4x }{ 3 }$$ ≤ $$\frac { 5 }{ 3 }$$, x∈ R.

Solution:

Given Inequation is -6 < -1 – $$\frac { 4x }{ 3 }$$ ≤ $$\frac { 5 }{ 3 }$$, x ∈ R.
-6 < -1 – $$\frac { 4x }{ 3 }$$ …….. (i)
-1 – $$\frac { 4x }{ 3 }$$ ≤ $$\frac { 5 }{ 3 }$$…………..(ii)
Firstly, solve the equation (i)
-6 < -1 – $$\frac { 4x }{ 3 }$$ …….. (i)
Now, add 1 on both sides of the above equation.
-6 + 1 < -1 + 1 – $$\frac { 4x }{ 3 }$$
-5 < – $$\frac { 4x }{ 3 }$$
Now, multiply the above equation with 3 into both sides.
-15 < – 4x
Now, divide the above equation with -4 into both sides. The inequality reverses on multiplying or dividing both sides by -1.
-15/-4 < – 4x/-4
x > 3.75
The value of x is greater than 3.75.
Now, solve the equation (ii)
-1 – $$\frac { 4x }{ 3 }$$ ≤ $$\frac { 5 }{ 3 }$$ …………..(ii)
Now, add 1 from both sides of the above equation.
-1 + 1 – $$\frac { 4x }{ 3 }$$ ≤ $$\frac { 5 }{ 3 }$$ + 1
– $$\frac { 4x }{ 3 }$$ ≤ $$\frac { 8 }{ 3 }$$
Now, multiply the above equation with 3 into both sides.
-4x ≤ 8
Now, divide the above equation with -4 into both sides. The inequality reverses on multiplying or dividing both sides by -1.
-4x/-4 ≤ 8/-4
x ≥ -2
The value of x is -2 or more than -2.

Therefore, the answer is -2 ≤ x < 3.75. Here least value of x is -2 and the greatest value of x is 3.75. So, the answer is {x : -2 ≤ x < 3.75 , x ∈ R}. The graph of the solution on a number line is

Question 10. Solve the given inequation and graph it on a number line 8x – 38 < $$\frac { 6x }{ 5 }$$ – 4 ≤ –$$\frac { 4 }{ 5 }$$ + 2x, x∈ R. Choose the correct answer from below options.
(i) 4 ≤ x < 5
(ii) -4 < x < 5
(iii) -4 ≤ x < 5
(iv) 4 ≤ x < -5

Solution:

Given Inequation is 8x – 38 < $$\frac { 6x }{ 5 }$$ – 4 ≤ –$$\frac { 4 }{ 5 }$$ + 2x, x∈ R.
8x – 38 < $$\frac { 6x }{ 5 }$$ – 4 …….. (i)
$$\frac { 6x }{ 5 }$$ – 4 ≤ –$$\frac { 4 }{ 5 }$$ + 2x…………..(ii)
Firstly, solve the equation (i)
8x – 38 < $$\frac { 6x }{ 5 }$$ – 4 …….. (i)
Now, solve the x terms. Move 14x to the left side of the above equation.
8x – $$\frac { 6x }{ 5 }$$ – 38 < – 4
$$\frac { 34x }{ 5 }$$ – 38 < -4
Now, add 38 on both sides of the above equation.
$$\frac { 34x }{ 5 }$$ – 38 + 38 < -4 + 38
$$\frac { 34x }{ 5 }$$ < 34
Now, multiply the above equation with 5 on both sides.
34x < 170
Now, divide the above equation with 34 into both sides.
34x/34 < 170/34
x < 5
The value of x is less than 5.
Now, solve the equation (ii)
$$\frac { 6x }{ 5 }$$ – 4 ≤ –$$\frac { 4 }{ 5 }$$ + 2x…………..(ii)
Now, solve the x terms. Move 14x to the left side of the above equation.
– 4 ≤ –$$\frac { 4 }{ 5 }$$ + 2x – $$\frac { 6x }{ 5 }$$
-4 ≤ –$$\frac { 4 }{ 5 }$$ + $$\frac { 4x }{ 5 }$$
Now, add $$\frac { 4 }{ 5 }$$ on both sides of the above equation.
-4 + $$\frac { 4 }{ 5 }$$ ≤ –$$\frac { 4 }{ 5 }$$ + $$\frac { 4 }{ 5 }$$ + $$\frac { 4x }{ 5 }$$
– $$\frac { 16 }{ 5 }$$ ≤ $$\frac { 4x }{ 5 }$$
Now, multiply the above equation with 5 on both sides.
-16 ≤ 4x
Now, divide the above equation with 4 into both sides. The inequality reverses on multiplying or dividing both sides by -1.
-16/4 ≤ 4x/4
-4 ≤ x
x ≥ -4
The value of x is -4 or more than -4.

Therefore, the answer is -4 ≤ x < 5. Here least value of x is -4 and the greatest value of x is 5. So, the answer is {x : -4 ≤ x < 5 , x ∈ R}. The graph of the solution on a number line is

The option (iii) -4 ≤ x < 5 is correct

Question 11. Solve the given inequation – $$\frac { 2x }{ 3 }$$ ≤ x – $$\frac { 8 }{ 3 }$$ < $$\frac { 1 }{ 3 }$$, x ∈ R.

Solution:

Given Inequation is – $$\frac { 2x }{ 3 }$$ ≤ x – $$\frac { 8 }{ 3 }$$ < $$\frac { 1 }{ 3 }$$, x ∈ R.
– $$\frac { 2x }{ 3 }$$ ≤ x – $$\frac { 8 }{ 3 }$$ …….. (i)
x – $$\frac { 8 }{ 3 }$$ < $$\frac { 1 }{ 3 }$$…………..(ii)
Firstly, solve the equation (i)
– $$\frac { 2x }{ 3 }$$ ≤ x – $$\frac { 8 }{ 3 }$$ …….. (i)
Now, solve the x terms. Move x to the left side of the above equation.
– x – $$\frac { 2x }{ 3 }$$ ≤ – $$\frac { 8 }{ 3 }$$
– $$\frac { 5x }{ 3 }$$ ≤ – $$\frac { 8 }{ 3 }$$
Now, multiply the above equation with -3 on both sides. The inequality reverses on multiplying or dividing both sides by -1.
5x ≥ 8
Now, divide the above equation with 5 into both sides.
5x/5 ≥ 8/5
x ≥ $$\frac { 8 }{ 5 }$$
The value of x is $$\frac { 8 }{ 5 }$$ or more than $$\frac { 8 }{ 5 }$$.
Now, solve the equation (ii)
x – $$\frac { 8 }{ 3 }$$ < $$\frac { 1 }{ 3 }$$…………..(ii)
Now, add $$\frac { 8 }{ 3 }$$ on both sides of the above equation.
x – $$\frac { 8 }{ 3 }$$ + $$\frac { 8 }{ 3 }$$ < $$\frac { 1 }{ 3 }$$ + $$\frac { 8 }{ 3 }$$
x < 3
The value of x is less than 3.

Therefore, the answer is $$\frac { 8 }{ 5 }$$ ≤ x < 3. Here least value of x is $$\frac { 8 }{ 5 }$$ and the greatest value of x is 3. So, the answer is {x : $$\frac { 8 }{ 5 }$$ ≤ x < 3 , x ∈ R}.

Question 12. Find the value of x which satisfies the inequation: –$$\frac { 17 }{ 3 }$$ < 1 – $$\frac { 4x }{ 3 }$$ ≤ 4, x ∈ W. Choose the correct answer from below options.
(i) -2.25 ≤ x < 5
(ii) 2.25 ≤ x < 5
(iii) -2.25 > x < 5
(iv) -2.25 < x < 5

Solution:

Given Inequation is –$$\frac { 17 }{ 3 }$$ < 1 – $$\frac { 4x }{ 3 }$$ ≤ 4, x ∈ W.
–$$\frac { 17 }{ 3 }$$ < 1 – $$\frac { 4x }{ 3 }$$ …….. (i)
1 – $$\frac { 4x }{ 3 }$$ ≤ 4…………..(ii)
Firstly, solve the equation (i)
–$$\frac { 17 }{ 3 }$$ < 1 – $$\frac { 4x }{ 3 }$$…….. (i)
Now, subtract 1 from both sides of the above equation.
–$$\frac { 17 }{ 3 }$$ -1 < 1 -1 – $$\frac { 4x }{ 3 }$$
–$$\frac { 20 }{ 3 }$$ < – $$\frac { 4x }{ 3 }$$
Now, multiply the above equation with -3 on both sides. The inequality reverses on multiplying or dividing both sides by -1.
20 > 4x
Now, divide the above equation with 4 into both sides.
20/4 > 4x/4
5 > x
The value of x is greater than 5.
Now, solve the equation (ii)
1 – $$\frac { 4x }{ 3 }$$ ≤ 4…………..(ii)
Now, subtract 1 on both sides of the above equation.
1 – 1 – $$\frac { 4x }{ 3 }$$ ≤ 4 – 1
– $$\frac { 4x }{ 3 }$$ ≤ 3
Now, multiply the above equation with -3 on both sides. The inequality reverses on multiplying or dividing both sides by -1.
4x ≥ -9
Now, divide the above equation with 4 into both sides.
4x/4 ≥ -9/4
x ≥ -9/4
x ≥ -2.25
The value of x is -2.25 or greater than -2.25.

Therefore, the answer is -2.25 ≤ x < 5. Here least value of x is -2.25 and the greatest value of x is 5. So, the answer is {x : -2.25 ≤ x < 5 , x ∈ W}. The option (i) -2.25 ≤ x < 5 is correct.

Question 13. Solve the given inequation 6 – 4x ≥ 2x – 24 where x ∈ N.

Solution:

Given Inequation is 6 – 4x ≥ 2x – 24 where x ∈ N
6 – 4x ≥ 2x – 24
Now, solve the x terms. Move 2x to the left side of the above equation.
6 – 4x – 2x ≥ – 24
6 – 6x ≥ – 24
Now, subtract 6 from both sides of the above equation.
6 – 6 – 6x ≥ – 24 – 6
– 6x ≥ – 30
Now, divide the above equation with -6 into both sides. The inequality reverses on multiplying or dividing both sides by -1.
– 6x/-6 ≥ – 30/-6
x ≤ -5
The value of x is -5 or less than 1.

Therefore, the answer is x ≤ -5 where x ∈ N.

Question 14. Solve the given inequation 24 + $$\frac { 11x }{ 3 }$$ ≤ 10 + 6x where x ∈ R. Choose the correct answer from below options.
(i) x ≤ 6
(ii) x ≥ 6
(iii) x < 6
(iv) x > 6

Solution:

Given Inequation is 24 + $$\frac { 11x }{ 3 }$$ ≤ 10 + 6x where x ∈ R.
24 + $$\frac { 11x }{ 3 }$$ ≤ 10 + 6x
Now, solve the x terms. Move $$\frac { 11x }{ 3 }$$ to the right side of the above equation.
24 ≤ 10 – $$\frac { 11x }{ 3 }$$ + 6x
24 ≤ 10 + $$\frac { 7x }{ 3 }$$
Now, subtract 10 from both sides of the above equation.
24 – 10 ≤ 10 – 10 + $$\frac { 7x }{ 3 }$$
14 ≤ $$\frac { 7x }{ 3 }$$
Now, multiply the above equation with 3 on both sides.
42 ≤ 7x
Now, divide the above equation with 7 into both sides.
42/7 ≤ 7x/7
6 ≤ x
x ≥ 6
The value of x is 6 or greater than 6.

Therefore, the answer is x ≥ 6 where x ∈ R. The option (ii) x ≥ 6 is correct.

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