Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable)

Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable)

Selina Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 4 Linear Inequations (in one variable)

Linear Inequations in One Variable Exercise 4A – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 1
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 2

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 \(\frac { a }{ c } <\frac { b }{ c }\)
(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:
(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 ⇒ \(\frac { a }{ c } >\frac { b }{ c }\)
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:
(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.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 3
Solution:
(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}
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 4
Since, the replacement set = W (set of whole numbers)
∴ Solution set = {0, 1, 2}
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 5
Since, the replacement set = W (set of whole numbers)
∴ Solution set = {0, 1}
(vi) 18 ≤ 3x – 2
18 + 2 ≤ 3x
20 ≤ 3x
x ≥ \(\frac { 20 }{ 3 }\)
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:
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:
25 – 4x ≤ 16
-4x ≤ 16 – 25
-4x ≤ -9
x ≥ \(\frac { 9 }{ 4 }\)
x ≥ 2.25
(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.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 6
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 7

Question 8.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 8
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 9
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:
2(x – 1) ≤ 9 – x
2x – 2 ≤ 9 – x
2x + x ≤ 9 + 2
3x ≤ 11
x ≤ \(\frac { 11 }{ 3 }\)
x ≤ 3.67
Since, x ∈ W, thus the required largest value of x is 3.

Question 10.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 10
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 11

Question 11.
Given x ∈ {integers}, find the solution set of:
-5 ≤ 2x – 3 < x + 2
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 12

Question 12.
Given x ∈ {whole numbers}, find the solution set of:
-1 ≤ 3 + 4x < 23

Solution:

Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 13

Linear Inequations in One Variable Exercise 4B – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 15
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 14
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 69

Question 2.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 16
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 17

Question 3.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 18
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 19

Question 4.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 20
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 21
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 22

Question 5.
x ∈ {real numbers} and -1 < 3 – 2x ≤ 7, evaluate x and represent it on a number line.
Solution:
-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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 23

Question 6.
List the elements of the solution set of the inequation
-3 < x – 2 ≤ 9 – 2x; x ∈ N.
Solution:
-3 < x – 2 ≤ 9 – 2x
-3 < x – 2 and x – 2 ≤ 9 – 2x
-1 < x and 3x ≤ 11
-1 < x ≤ \(\frac { 11 }{ 3 }\)
Since, x ∈ N
∴ Solution set = {1, 2, 3}

Question 7.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 24
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 25

Question 8.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 26
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 27
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:
-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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 28

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:
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:
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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 29.

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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 70

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:
(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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 31
(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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 32

Question 14.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 33
Solution:
(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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 34
B’ = {x ∈ R: 3 < x ≤ -4}
A ∩ B’ = {x ∈ R: 3 ≤ x < 5}
It can be represented on number line as:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 35

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:
(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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 36
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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 37
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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 38
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:
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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 39

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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 40

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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 41

Question 19.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 42
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 43

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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 44

Question 21.
Solve the following inequation and represent the solution set on the number line 2x – 5 ≤ 5x +4 < 11, where x ∈ I
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 45

Question 22.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 46
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 47

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:
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}
A ∩ B = {x: x ≥ 4, x ∈ R}
It can be represented on number line as:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 48

Question 24.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 49
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 50

Question 25.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 51
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 52

Question 26.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 53
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 54

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:
Let the required integers be x, x + 1 and x + 2.
According to the given statement,
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 55
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.
2y – 3 < y + 1 ≤ 4y + 7, y ∈ R
Solution:
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:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 56

Question 29.
Solve the inequation:
3z – 5 ≤ z + 3 < 5z – 9, z ∈ R.
Graph the solution set on the number line.
Solution:
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, x ∈ R }
It can be represented on a number line as:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 57

Question 30.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 58
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 59

Question 31.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 60
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) q32

Question 32.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 61
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 62
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 63

Question 33.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 64
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 65
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 66

Question 34.
Solve the following in equation and write the solution set:
13x – 5 < 15x + 4 < 7x + 12, x ∈ R
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 67
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) - 68

Question 35.
Solve the following inequation, write the solution set and represent it on the number line.
-3(x – 7) ≥ 15 – 7x > x+1/3, x R.
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) q36

Question 36.
Solve the following inequation and represent the solution set on a number line.
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) q36
Solution:
Selina Concise Mathematics Class 10 ICSE Solutions Linear Inequations (in one variable) q37

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