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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.