SELINA Solutions for Class 10 Maths Chapter 4 - Linear Inequations (in one variable)
Revise Selina Solutions for ICSE Class 10 Mathematics Chapter 4 Linear Inequations (in One Variable) for exam preparation. Learn to solve linear inequations algebraically and graphically with the step-wise solutions created by Maths experts. These solutions are prepared according to the latest syllabus and exam guidelines.
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Chapter 4 - Linear Equations in One Variable Exercise Ex. 4(A)
State, true or false:
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.
(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.
If x N, find the solution set of inequations.
(i) 5x + 3 2x + 18
(ii) 3x - 2 < 19 - 4x
(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}.
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
(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, …}
Solve the inequation:
3 - 2x x - 12 given that x
N.
3 - 2x x - 12
-2x - x -12 - 3
-3x -15
x 5
Since, x N, therefore,
Solution set = {1, 2, 3, 4, 5}
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.
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.
If the replacement set is the set of real numbers, solve:
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}
Find the smallest value of x for which 5 - 2x <, where x is an integer.
Thus, the required smallest value of x is -1.
Find the largest value of x for which
2(x - 1) 9 - x and x
W.
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.
Solve the inequation: and x
R.
Solution set = {x: x
R and x
6}
Given x {integers}, find the solution set of:
Since, x {integers}
Solution set = {-1, 0, 1, 2, 3, 4}
Given x {whole numbers}, find the solution set of:
.
Since, x {whole numbers}
Solution set = {0, 1, 2, 3, 4}
Chapter 4 - Linear Equations in One Variable Exercise Ex. 4(B)
Represent the following inequalities on real number lines:
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:
For each graph given, write an inequation taking x as the variable:
For the following inequations, graph the solution set on the real number line:
The solution set on the real number line is:
The solution set on the real number line is:
Represent the solution of each of the following inequalities on the real number line:
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:
x {real numbers} and -1 < 3 - 2x
7, evaluate x
and represent it on a number line.
-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:
List the elements of the solution set of the inequation
-3 < x - 2 9 - 2x; x
N.
-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}
Find the range of values of x which satisfies
Graph these values of x on the number line.
-3
x and x < 3
-3
x < 3
The required graph of the solution set is:
Find the values of x, which satisfy the inequation:
Graph the solution on the number line.
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
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.
-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:
If 5x - 3 5 + 3x
4x + 2, express it as a
x
b and then state the values of a
and b.
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.
Solve the following inequation and graph the solution set on the number line:
2x - 3 < x + 2 3x + 5, x
R.
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:
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
(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
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
(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:
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.
(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:
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
(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
Illustrate the set {x: -3 x < 0 or x > 2, x
R} on the real number line.
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:
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
(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:
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.
and
(i)
(ii) P - Q = {x: 1 < x < 5, x R}
(iii) {x: 1 < x < 5, x
R}
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
(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
.
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.
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}
Solve the following inequation and represent the solution set on the number line 2x - 5 ≤ 5x +4 < 11, where xI
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
Given that x I, solve the inequation and graph
the solution on the number line:
Solution set = {5, 6}
It can be graphed on number line as:
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.
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:
Find the set of values of x, satisfying:
7x + 3 3x - 5 and
, where x
N.
7x + 3 3x - 5
4x -8
x -2
Since, x N
Solution set = {1, 2, 3, 4, 5}
Solve:
(i) , where x is a positive odd integer.
(ii) , where x is a positive even integer.
(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}
Solve the inequation:
, x
W. Graph the solution set on the number line.
Since, x W
Solution set = {0, 1, 2}
The solution set can be represented on number line as:
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.
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.
Solve the given inequation and graph the solution on the number line.
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:
Solve the inequation:
3z - 5 z + 3 < 5z - 9, z
R.
Graph the solution set on the number line.
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:
Solve the following inequation and represent the solution set on the number line.
-3 < R
The solution set can be represented on a number line as:
Solve the following inequation and represent the solution set on the number line:
Consider the given inequation:
⇒ -4 ≤ x < 5; where x ∊ R
The solution set can be represented on a number line as follows:
Solve the following in equation, write the solution set and represent it on the number line:
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.
Solve the following inequation, write the solution set and represent it on the number line.
The solution set is represented on number line as follows:
Solve the following inequation and represent the solution set on a number line.
As,
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