Chapter 4 - Linear Equations in One Variable Exercise Ex. 4(B)

Question 1

Represent the following inequalities on real number lines:

Solution 1

Solution on number line is:

 

Solution on number line is:

 

Solution on number line is:

 

 

Solution on number line is:

 

 

Solution on number line is:

 

 

Solution on number line is:

 

 

Solution on number line is:

 

 

 

Question 2

For each graph given, write an inequation taking x as the variable:

Solution 2

Question 3

For the following inequations, graph the solution set on the real number line:

Solution 3

The solution set on the real number line is:

The solution set on the real number line is:

Question 4

Represent the solution of each of the following inequalities on the real number line:

Solution 4

The solution on number line is as follows:

The solution on number line is as follows:

 

The solution on number line is as follows:

 

The solution on number line is as follows:

 

The solution on number line is:

 

The solution on number line is:

Question 5

x  {real numbers} and -1 < 3 - 2x  7, evaluate x and represent it on a number line.

Solution 5

-1 < 3 - 2x  7

-1 < 3 - 2x and 3 - 2x  7

2x < 4 and -2x  4

x < 2 and x  -2

Solution set = {-2  x < 2, x  R}

Thus, the solution can be represented on a number line as:

Question 6

List the elements of the solution set of the inequation

-3 < x - 2  9 - 2x; x  N.

Solution 6

-3 < x - 2  9 - 2x

-3 < x - 2 and x - 2  9 - 2x

-1 < x and 3x  11

-1 < x  

Since, x  N

Solution set = {1, 2, 3}

Question 7

Find the range of values of x which satisfies

Graph these values of x on the number line.

Solution 7

 -3  x and x < 3

 -3  x < 3

The required graph of the solution set is:

Question 8

Find the values of x, which satisfy the inequation:

Graph the solution on the number line.

Solution 8

Thus, the solution set is {x ∊ N: -2 ≤ x ≤3.75}

Since x ∊ N, the values of x are 1, 2, 3

The solution on number line is given by

Question 9

Given x  {real numbers}, find the range of values of x for which -5  2x - 3 < x + 2 and represent it on a number line.

Solution 9

-5  2x - 3 < x + 2

-5  2x - 3 and 2x - 3 < x + 2

-2  2x and x < 5

-1  x and x < 5

Required range is -1  x < 5.

The required graph is:

Question 10

If 5x - 3  5 + 3x  4x + 2, express it as a  x  b and then state the values of a and b.

Solution 10

5x - 3  5 + 3x  4x + 2

5x - 3  5 + 3x and 5 + 3x  4x + 2

2x  8 and -x  -3

x  4 and x  3

Thus, 3  x  4.

Hence, a = 3 and b = 4.

Question 11

Solve the following inequation and graph the solution set on the number line:

2x - 3 < x + 2  3x + 5, x  R.

Solution 11

2x - 3 < x + 2  3x + 5

2x - 3 < x + 2 and x + 2  3x + 5

x < 5 and -3  2x

x < 5 and -1.5  x

Solution set = {-1.5  x < 5}

The solution set can be graphed on the number line as:

Question 12

Solve and graph the solution set of:

(i) 2x - 9 < 7 and 3x + 9  25, x  R

(ii) 2x - 9  7 and 3x + 9 > 25, x  I

(iii) x + 5  4(x - 1) and 3 - 2x < -7, x  R

Solution 12

(i) 2x - 9 < 7 and 3x + 9  25

2x < 16 and 3x  16

x < 8 and x 5

Solution set = { x  5, x  R}

The required graph on number line is:

(ii) 2x - 9  7 and 3x + 9 > 25

2x  16 and 3x > 16

x  8 and x > 5

Solution set = {5 < x  8, x  I} = {6, 7, 8}

The required graph on number line is:

(iii) x + 5  4(x - 1) and 3 - 2x < -7

9  3x and -2x < -10

3  x and x > 5

Solution set = Empty set

Question 13

Solve and graph the solution set of:

(i) 3x - 2 > 19 or 3 - 2x  -7, x  R

(ii) 5 > p - 1 > 2 or 7  2p - 1  17, p  R

Solution 13

(i) 3x - 2 > 19 or 3 - 2x  -7

3x > 21 or -2x  -10

x > 7 or x  5

Graph of solution set of x > 7 or x  5 = Graph of points which belong to x > 7 or x  5 or both.

Thus, the graph of the solution set is:

(ii) 5 > p - 1 > 2 or 7  2p - 1  17

6 > p > 3 or 8  2p  18

6 > p > 3 or 4  p  9

Graph of solution set of 6 > p > 3 or 4  p  9

= Graph of points which belong to 6 > p > 3 or 4  p  9 or both

= Graph of points which belong to 3 < p  9

Thus, the graph of the solution set is:

Question 14

The diagram represents two inequations A and B on real number lines:

(i) Write down A and B in set builder notation.

(ii) Represent A  B and A  B' on two different number lines.

Solution 14

(i) A = {x  R: -2  x < 5}

 

B = {x  R: -4  x < 3}

 

(ii) A  B = {x  R: -2  x < 5}

It can be represented on number line as:

 

 

B' = {x  R: 3 < x  -4}

 

A  B' = {x  R: 3  x < 5}

 

It can be represented on number line as:

 

 

Question 15

Use real number line to find the range of values of x for which:

(i) x > 3 and 0 < x < 6

(ii) x < 0 and -3  x < 1

(iii) -1 < x  6 and -2  x  3

Solution 15

(i) x > 3 and 0 < x < 6

Both the given inequations are true in the range where their graphs on the real number lines overlap.

The graphs of the given inequations can be drawn as:

x > 3

0 < x < 6

From both graphs, it is clear that their common range is

3 < x < 6

(ii) x < 0 and -3  x < 1

Both the given inequations are true in the range where their graphs on the real number lines overlap.

The graphs of the given inequations can be drawn as:

x < 0

-3 x < 1

From both graphs, it is clear that their common range is

-3  x < 0

(iii) -1 < x  6 and -2  x  3

Both the given inequations are true in the range where their graphs on the real number lines overlap.

The graphs of the given inequations can be drawn as:

-1 < x 6

-2  x  3

From both graphs, it is clear that their common range is

-1 < x  3

Question 16

Illustrate the set {x: -3  x < 0 or x > 2, x  R} on the real number line.

Solution 16

Graph of solution set of -3  x < 0 or x > 2

= Graph of points which belong to -3  x < 0 or x > 2 or both

Thus, the required graph is:

Question 17

Given A = {x: -1 < x  5, x  R} and B = {x: -4  x < 3, x  R}

Represent on different number lines:

(i) A  B

(ii) A'  B

(iii) A - B

Solution 17

(i) A  B = {x: -1 < x < 3, x  R}

It can be represented on a number line as:

 

(ii) Numbers which belong to B but do not belong to A' = B - A

A'  B = {x: -4  x  -1, x  R}

It can be represented on a number line as:

 

(iii) A - B = {x: 3  x  5, x  R}

It can be represented on a number line as:

Question 18

P is the solution set of 7x - 2 > 4x + 1 and Q is the solution set of 9x - 45  5(x - 5); where x  R. Represent:

(i) P  Q

(ii) P - Q

(iii) P  Q'

on different number lines.

Solution 18

and

(i) 

(ii) P - Q = {x: 1 < x < 5, x  R}

(iii) {x: 1 < x < 5, x  R}

Question 19

Solution 19

Question 20

Find the range of values of x, which satisfy:

Graph, in each of the following cases, the values of x on the different real number lines:

(i) x  W (ii) x  Z (iii) x  R

Solution 20

(i)If x  W, range of values of x is {0, 1, 2, 3, 4, 5, 6}.

 

 

(ii) If x  Z, range of values of x is {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

 

 

(iii)If x  R, range of values of x is .

 

Question 21

Given: A = {x: -8 < 5x + 2  17, x  I}, B = {x: -2  7 + 3x < 17, x  R}

Where R = {real numbers} and I = {integers}. Represent A and B on two different number lines. Write down the elements of A  B

Solution 21

A = {x: -8 < 5x + 2  17, x  I}

= {x: -10 < 5x  15, x  I}

= {x: -2 < x  3, x  I}

It can be represented on number line as follows:

 

 

B = {x: -2  7 + 3x < 17, x  R}

= {x: -9  3x < 10, x  R}

= {x: -3  x < 3.33, x  R}

It can be represented on number line as follows:

A  B = {-1, 0, 1, 2, 3}

Question 22

Solve the following inequation and represent the solution set on the number line 2x - 5 ≤ 5x +4 < 11, where xI

Solution 22

2x - 5 ≤ 5x + 4 and 5x +4 < 11

2x - 9 ≤ 5x  and 5x < 11 - 4

  -9 ≤ 3x and 5x < 7

x  - 3 and x < 

x  - 3 and x < 

 

Since x I, the solution set is 

And the number line representation is

 

Question 23

Given that x  I, solve the inequation and graph the solution on the number line:

Solution 23

Solution set = {5, 6}

It can be graphed on number line as:

Question 24

Given:

A = {x: 11x - 5 > 7x + 3, x  R} and

B = {x: 18x - 9  15 + 12x, x  R}.

Find the range of set A  B and represent it on number line.

Solution 24

A = {x: 11x - 5 > 7x + 3, x  R}

= {x: 4x > 8, x  R}

= {x: x > 2, x  R}

B = {x: 18x - 9  15 + 12x, x  R}

= {x: 6x  24, x  R}

= {x: x  4, x  R}

Range of A  B = {x: x  4, x  R}

It can be represented on number line as:

 

Question 25

Find the set of values of x, satisfying:

7x + 3  3x - 5 and , where x  N.

Solution 25

7x + 3 3x - 5

4x  -8

x  -2

Since, x  N

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

Question 26

Solve:

(i) , where x is a positive odd integer.

(ii) , where x is a positive even integer.

Solution 26

(i) 

Since, x is a positive odd integer

Solution set = {1, 3, 5}

(ii) 

Since, x is a positive even integer

Solution set = {2, 4, 6, 8, 10, 12, 14}

Question 27

Solve the inequation:

, x W. Graph the solution set on the number line.

Solution 27 

Since, x  W

Solution set = {0, 1, 2}

The solution set can be represented on number line as:

 

Question 28

Find three consecutive largest positive integers such that the sum of one-third of first, one-fourth of second and one-fifth of third is atmost 20.

Solution 28

Let the required integers be x, x + 1 and x + 2.

According to the given statement,

Thus, the largest value of the positive integer x is 24.

Hence, the required integers are 24, 25 and 26.

Question 29

Solve the given inequation and graph the solution on the number line.

Solution 29

2y - 3 < y + 1  4y + 7, y R

2y - 3 - y < y + 1 - y  4y + 7 - y

 y - 3 < 1  3y + 7

 y - 3 < 1 and 1 3y + 7

 y < 4 and 3y  - 6  y  - 2

 - 2  y < 4

The graph of the given equation can be represented on a number line as:

Question 30

Solve the inequation:

3z - 5  z + 3 < 5z - 9, z  R.

Graph the solution set on the number line.

Solution 30

3z - 5  z + 3 < 5z - 9

3z - 5  z + 3 and z + 3 < 5z - 9

2z  8 and 12 < 4z

z  4 and 3 < z

Since, z  R

Solution set = {3 < z  4, Z  R }

It can be represented on a number line as:

 

Question 31

Solve the following inequation and represent the solution set on the number line.

-3 < R

Solution 31

The solution set can be represented on a number line as:

Question 32

Solve the following inequation and represent the solution set on the number line:

Solution 32

Consider the given inequation:

⇒ -4 ≤ x < 5; where x ∊ R

The solution set can be represented on a number line as follows:

 

Question 33

Solve the following in equation, write the solution set and represent it on the number line:

Solution 33

 

 

 

 

Question 34

Solution 34

 

 

Question 35

Solve the following in equation and write the solution set:

13x - 5 < 15x + 4 < 7x + 12, x ∈ R

Represent the solution on a real number line.

Solution 35

Question 36

Solve the following inequation, write the solution set and represent it on the number line.

Solution 36

 

The solution set is represented on number line as follows:

Question 37

Solve the following inequation and represent the solution set on a number line.

Solution 37

As,