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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.