**1. Solve the inequation, 3x – 11 < 3 where x ∈ {1,2,3,……,10}. Also, represent its solution on a number line.**

**Solution:**

We have given that

3x – 11 <3

3x < 3 + 11

3x <14

∴x <14/3

But, x ∈ {1,2,3,……,10}

Hence, the solution set is {1,2,3,4}.

The solution is representing on number line:

**2. Solve 2(x – 3) < 1,x ∈ {1,2,3,….10}**

**Solution:**

We have given that,

2(x – 3) < 1

2x – 6 < 1

2x < 7

**3. Solve 5 – 4x > 2 – 3x,x ∈ W. Also represent its solution on the number line.**

**Solution:**

We have given that,

5 – 4x > 2 – 3x

– 4x + 3x > 2 – 5

-x > -3

On multiplying both sides by -1, the inequality reverses

∴x < 3

But, x ∈ W

The solution set is {0,1,2}

The solution is representing on number line:

**4. List the solution set of 30 – 4 (2x – 1) < 30, given that x is a positive integer.**

**Solution:**

We have given that,

30 – 4 (2x – 1) < 30

30 – 8x + 4 < 30

34 – 8x < 30

-8x < 30 – 34

-8x < -4

On multiplying both sides by -1, the inequality reverses

8x > 4

Hence, the solution set is {3,4,5,6,7,8}.

**8. Solve x – 3 (2 + x) > 2 (3x – 1),x ∈ { – 3,– 2,– 1,0,1,2,3}. Also represent its solution on the number line.**

**Solution:**

**9. Given x ∈ {1,2,3,4,5,6,7,9} solve x – 3 < 2x – 1.**

**Solution:**

We have given that,

x – 3 < 2x – 1

x – 2x < – 1 + 3

-x < 2

∴ x > -2

But, x ∈ {1,2,3,4,5,6,7,9}

Hence, the solution set is {1,2,3,4,5,6,7,9}.

**10. Given A = {x: x ∈ I,– 4 ≤ x ≤ 4}, solve 2x – 3 < 3 where x has the domain A. Graph the solution set on the number line.**

**Solution:**

We have given that,

2x – 3 < 3

2x< 3+3

2x < 6

∴ x < 3

But x has the domain A = {x: x ∈ I,– 4 ≤ x ≤ 4}

A = {-4,-3,-2,-1,0,1,2,3,4}

Hence, the solution set is {-4,-3,-2,-1,0,1,2}.

The solution is representing on number line:

**Solution:**

(i) x is a positive odd integer

We have given that,

8x + 12 ≥ 9x – 3

-9x + 8x ≥ -12 – 3

-x ≥ -15

∴ x ≤ 15

As x is positive even integer.

Hence, the solution set is {2,4,6,8,10,12,14}.

**17. Given x ∈ {1,2,3,4,5,6,7,9}, find the values of x for which -3 < 2x – 1 < x + 4.**

**Solution:**

We have given that,

-3 < 2x – 1 < x + 4

Now, we have

-3 < 2x – 1 and 2x – 1 < x + 4

-2x < 3 – 1 and 2x – x < 4 + 1

-2x < 2 and x < 5

∴x > -1 and x < 5

As x ∈ {1,2,3,4,5,6,7,9}

Thus, The solution set is {1,2,3,4}.

**18. Solve: 1 ≥ 15 – 7x > 2x – 27,x ∈ N**

**Solution:**

We have given that,

1 ≥ 15 – 7x > 2x – 27,

Now, we have

1 ≥ 15 – 7x and 15 – 7x > 2x – 27

7x ≥ 15 – 1 and -2x – 7x > -27 – 15

7x ≥ 14 and -9x > -42

**19. If x ∈ Z,solve 2 + 4x < 2x – 5 ≤ 3x. Also represent its solution on the number line.**

**Solution:**

We have given that,

2 + 4x < 2x – 5 ≤ 3x

Now, we have

2 + 4x < 2x – 5 and 2x – 5 ≤ 3x

4x – 2x < -5 – 2 and 2x – 3x ≤ 5

2x < -7 and -x ≤ 5

72 + 11x ≤ 30 + 18x

11x – 18x ≤ 30 – 72

-7x ≤ -42

-x ≤ -6

∴x ≥ 6

As x ∈ R

Thus, The solution set is {x∶ x ∈ R,x ≥ 6}

The solution is representing on number line:

**23. Solve the inequation – 3 ≤ 3 – 2x < 9,x ∈ R. Represent your solution on a number line.**

**Solution:**

We have given that,

– 3 ≤ 3 – 2x < 9

– 3 – 3 ≤ – 2x < 9 – 3

-6 ≤ -2x < 6

-3 ≤ -x < 3

∴-3 < x ≤ 3

As x ∈ R

The solution set is {x: x ∈ R,-3 < x ≤ 3}

The solution is representing on number line:

**24. Solve 2 ≤ 2x – 3 ≤ 5,x ∈ R and mark it on a number line.**

**Solution:**

We have given that,

2 ≤ 2x – 3 ≤ 5

2 + 3 ≤ 2x ≤ 5 + 3

5 ≤ 2x ≤ 8

**25. Given that x ∈ R, solve the following inequation and graph the solution on the number line: -1 ≤ 3 + 4x < 23.**

**Solution:**

We have given that,

-1 ≤ 3 + 4x < 23

-1 – 3 ≤ 4x < 23 – 3

-4 ≤ 4x < 20

**32. If x ∈ I,A is the solution set of 2 (x – 1) < 3x – 1 and B is the solution set of 4x – 3 ≤ 8 + x, find A ∩ B.**

**Solution:**

We have given that,

2 (x – 1)< 3x – 1 and 4x – 3 ≤ 8 + x for x ∈ I

Solving for both, we have

2x – 3x < 2 – 1 and 4x – x ≤ 8 + 3

**33. If P is the solution set of -3x + 4 < 2x – 3,x ∈ N and Q is the solution set of 4x – 5 < 12,x ∈ W, find**

**(i) P ∩ Q**

**(ii) Q – P.**

**(ii) Q – P.**

**Solution:**

We have given that,

-3x + 4 < 2x – 3 where x ∈ N and 4x – 5 < 12 where x ∈ W

So, solving

-3x + 4 < 2x – 3 where x ∈ N

-3x – 2x < -3 – 4

-5x < -7

Hence, B = {x : x ≥ 4, x ∈ R}

Thus, A ∩ B = x ≥ 4

The solution is representing on number line:

**35. Given: P {x∶ 5 < 2x – 1 ≤ 11,x ∈ R}**

**Q {x∶ – 1 ≤ 3 + 4x < 23,x ∈ I} where**

**R = (real numbers), I = (integers)**

**Represent P and Q on number line. Write down the elements of P ∩ Q.**

**Solution:**

Given, P {x∶ 5 < 2x – 1 ≤ 11,x ∈ R} and Q {x∶ – 1 ≤ 3 + 4x < 23,x ∈ I}

Solving for P,

5 < 2x – 1 ≤ 11

5 + 1 < 2x ≤ 11 + 1

6 < 2x ≤ 12

∴3 < x ≤ 6

Hence, P = P {x∶ 3 < x ≤ 6,x ∈ R}

The solution is representing on number line:

2x > -3

∴x > -3/2

Hence, for x ∈ I the smallest value of x is -1.

**37. Given 20 – 5 x < 5 (x + 8), find the smallest value of x, when**

**(i) x ∈ I**

**(ii) x ∈ W**

**(iii) x ∈ N.**

**Solution:**

We have given that,

20 – 5 x < 5 (x + 8)

20 – 5x < 5x + 40

-5x – 5x < 40 – 20

-10x < 20

Thus,

(i) For x ∈ I, the smallest value = -1

(ii) For x ∈ W, the smallest value = 0

(iii) For x ∈ N, the smallest value = 1

**39. Solve the given inequation and graph the solution on the number line:**

**2y – 3 < y + 1 ≤ 4y + 7;y ∈ R.**

**Solution:**

We have given that,

2y – 3 < y + 1 ≤ 4y + 7

Now, we have

2y – 3 < y + 1 and y + 1 ≤ 4y + 7

2y – y < 1 + 3 and y – 4y ≤ 7 – 1

y < 4 and -3y ≤ 6

y < 4 and -y ≤ 2 ⇒ y ≥ -2

∴ -2 ≤ y < 4

The solution set is {y : -2 ≤ y < 4, y ∈ R}

The solution is representing on number line:

x ≥ -3 and x ≤ 4

∴ -3 ≤ x ≤ 4

Thus, the solution set is {-3, -2, -1, 0, 1, 2, 3, 4}

The solution is representing on number line:

**41. Find the greatest integer which is such that if 7 is added to its double, the resulting number becomes greater than three times the integer.**

**Solution**:

Let’s consider the greatest integer to be x

Then according to the given condition,

we have

2x + 7 > 3x

2x – 3x > -7

-x > -7

∴x < 7 ,x ∈ R

Hence, the greatest integer value is 6.

**42. One-third of a bamboo pole is buried in mud, one-sixth of it is in water and the part above the water is greater than or equal to 3 metres. Find the length of the shortest pole.**

**Solution:**

Let’s assume the length of the shortest pole = x metre

Now,

Multiplying by 6

Therefore, the length of the shortest pole is 6 metres.

**CHAPTER TEST**

**1. Solve the inequation: 5x – 2 ≤ 3 (3 – x) where x ∈ {-2, -1, 0, 1, 2, 3, 4}. Also represent its solution on the number line.**

**Solution:**

We have given that,

5x – 2 ≤ 3 (3 – x)

5x – 2 ≤ 9 – 3x

5x + 3x ≤ 9 + 2

8x ≤ 11

**2. Solve the inequation: 6x – 5 < 3x + 4,x ∈ I**

**Solution:**

We have given that,

6x – 5 < 3x + 4

6x – 3x < 4 + 5

3x < 9

x <9/3

∴x < 3

As x ∈ I

Hence, the solution set is {2, 1, 0, -1, -2, …}

**3. Find the solution set of the inequation x + 5 ≤ 2x + 3; x ∈ R**

**Graph the solution set on the number line.**

**Solution:**

We have given that,

x + 5 ≤ 2x + 3

x – 2x ≤ 3 – 5

-x ≤ -2

∴x ≥ 2

As x ∈ R

Hence, the solution set is {2, 3, 4, 5, …}

The solution is representing on number line:

**4. If x ∈ R (real numbers) and -1 < 3 – 2x ≤ 7, find solution set and present it on a number line.**

**Solution:**

We have given that,

-1 < 3 – 2x ≤ 7

-1 – 3 < -2x ≤ 7 – 3

-4 < -2x ≤ 4