SELINA Solutions for Class 10 Maths Chapter 4 - Linear Inequations (in one variable)
Chapter 4 - Linear Equations in One Variable Exercise Ex. 4(A)
Question 1
State, true or false:
Solution 1
Question 2
State, whether the following statements are true or false:
(i) a < b, then a - c < b - c
(ii) If a > b, then a + c > b + c
(iii) If a < b, then ac > bc
(iv) If a > b,
then 
(v) If a - c > b - d, then a + d > b + c
(vi) If a < b, and c > 0, then a - c > b - c
Where a, b, c and d
are real numbers and c 
0.
Solution 2
(i) a < b 
a - c < b - c
The given statement is true.
(ii) If a > b a + c > b + c
The given statement is true.
(iii) If a < b ac < bc
The given statement is false.
(iv) If a > b 
The given statement is false.
(v) If a - c > b
- d a + d > b + c
The given statement is true.
(vi) If a < b a - c < b - c (Since, c > 0)
The given statement is false.
Question 3
If x 
N, find the solution set of inequations.
(i) 5x + 3 
2x + 18
(ii) 3x - 2 < 19 - 4x
Solution 3
(i) 5x + 3 2x + 18
5x - 2x 18 - 3
3x 15
x 5
Since, x N, therefore solution set is {1, 2, 3, 4, 5}.
(ii) 3x - 2 < 19 - 4x
3x + 4x < 19 + 2
7x < 21
x < 3
Since, x N, therefore solution set is {1, 2}.
Question 4
If the replacement set is the set of whole numbers, solve:
(i) x + 7 
11
(ii) 3x - 1 > 8
(iii) 8 - x > 5
(iv) 7 - 3x 
(v) 
(vi) 18 3x - 2
Solution 4
(i) x + 7 11
x 11 - 7
x 4
Since, the replacement set = W (set of whole numbers)
Solution set = {0, 1, 2, 3, 4}
(ii) 3x - 1 > 8
3x > 8 + 1
x > 3
Since, the replacement set = W (set of whole numbers)
Solution set = {4, 5, 6, …}
(iii) 8 - x > 5
- x > 5 - 8
- x > -3
x < 3
Since, the replacement set = W (set of whole numbers)
Solution set = {0, 1, 2}
(iv) 7 - 3x
-3x - 7
-3x 
x 
Since, the replacement set = W (set of whole numbers)
Solution set = {0, 1, 2}
(v)
Since, the replacement set = W (set of whole numbers)
Solution set = {0, 1}
(vi) 18 3x - 2
18 + 2 3x
20 3x
Since, the replacement set = W (set of whole numbers)
Solution set = {7, 8, 9, …}
Question 5
Solve the inequation:
3 - 2x 
x - 12 given that x 
N.
Solution 5
3 - 2x x - 12
-2x - x -12 - 3
-3x -15
x 
5
Since, x N, therefore,
Solution set = {1, 2, 3, 4, 5}
Question 6
If 25 - 4x 
16, find:
(i) the smallest value of x, when x is a real number,
(ii) the smallest value of x, when x is an integer.
Solution 6
25 - 4x 16
-4x 16 - 25
-4x -9
x 
x 
(i) The smallest value of x, when x is a real number, is 2.25.
(ii) The smallest value of x, when x is an integer, is 3.
Question 7
If the replacement set is the set of real numbers, solve:
Solution 7
Since, the replacement set of real numbers.
Solution set = {x: x 
R and 
}
Since, the replacement set of real numbers.
Solution set = { x: x R and 
}
Since, the replacement set of real numbers.
Solution set = { x: x R and x > 80}
Since, the replacement set of real numbers.
Solution set = { x: x R and x > 13}
Question 8
Find the smallest value of x for which 5 - 2x <
, where x is an integer.
Solution 8
Thus, the required smallest value of x is -1.
Question 9
Find the largest value of x for which
2(x - 1) 
9 - x and x 
W.
Solution 9
2(x - 1) 9 - x
2x - 2 9 - x
2x + x 9 + 2
3x 11
Since, x W, thus the required largest value of x is 3.
Question 10
Solve the inequation: 
and x 
R.
Solution 10
Solution set = {x: x 
R and x 
6}
Question 11
Given x 
{integers}, find the solution set of:
Solution 11
Since, x {integers}
Solution set = {-1, 0, 1, 2, 3, 4}
Question 12
Given x 
{whole numbers}, find the solution set of:
.
Solution 12
Since, x {whole numbers}
Solution set = {0, 1, 2, 3, 4}
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
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 19
(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 20
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 20
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 21
Solve the following inequation and represent the solution set on the number line 2x - 5 ≤ 5x +4 < 11, where x
I
Solution 21
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 22
Given that x 
I, solve the inequation and graph
the solution on the number line:
Solution 22
Solution set = {5, 6}
It can be graphed on number line as:
Question 23
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 23
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 24
Find the set of values of x, satisfying:
7x + 3 
3x - 5 and 
, where x 
N.
Solution 24
7x + 3 3x - 5
4x -8
x -2
Since, x N
Solution set = {1, 2, 3, 4, 5}
Question 25
Solve:
(i) 
, where x is a positive odd integer.
(ii) 
, where x is a positive even integer.
Solution 25
(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 26
Solve the inequation:
, x 
W. Graph the solution set on the number line.
Solution 26
Since, x 
W
Solution set = {0, 1, 2}
The solution set can be represented on number line as:
Question 27
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 27
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 28
Solve the given inequation and graph the solution on the number line.
Solution 28
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 29
Solve the inequation:
3z - 5 
z + 3 < 5z - 9, z 
R.
Graph the solution set on the number line.
Solution 29
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 30
Solve the following inequation and represent the solution set on the number line.
-3 < 
R
Solution 30
The solution set can be represented on a number line as:
Question 31
Solve the following inequation and represent the solution set on the number line:
Solution 31
Consider the given inequation:
⇒ -4 ≤ x < 5; where x ∊ R
The solution set can be represented on a number line as follows:
Question 32
Solve the following in equation, write the solution set and represent it on the number line:
Solution 32
Question 33
Solution 33
Question 34
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 34
Question 35
Solve the following inequation, write the solution set and represent it on the number line.
Solution 35
The solution set is represented on number line as follows:
Question 36
Solve the following inequation and represent the solution set on a number line.
Solution 36
As, 
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