NCERT Textbook - Linear Inequalities | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


5 Chapter
vMathematics is the art of saying many things in many
different ways. – MAXWELLv
5.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by translating them in the form of equations. Now a
natural question arises: ‘Is it always possible to translate a statement problem in the
form of an equation? For example, the height of all the students in your class is less
than 160 cm. Your classroom can occupy atmost 60 tables or chairs or both. Here we
get certain statements involving a sign ‘<’ (less than), ‘>’ (greater than), ‘=’ (less than
or equal) and = (greater than or equal) which are known as inequalities.
In this Chapter, we will study linear inequalities in one and two variables. The
study of inequalities is very useful in solving problems in the field of science, mathematics,
statistics, economics, psychology, etc.
5.2 Inequalities
Let us consider the following situations:
(i) Ravi goes to market with ` 200 to buy rice, which is available in packets of 1kg. The
price of one packet of rice is ` 30. If x denotes the number of packets of rice, which he
buys, then the total amount spent by him is ` 30x. Since, he has to buy rice in packets
only, he may not be able to spend the entire amount of ` 200. (Why?) Hence
30x < 200 ... (1)
Clearly the statement (i) is not an equation as it does not involve the sign of equality.
(ii) Reshma has ` 120 and wants to buy some registers and pens. The cost of one
register is  ` 40 and that of a pen is ` 20. In this case, if x denotes the number of
registers and y, the number of pens which Reshma buys, then the total amount spent by
her is ` (40x  +  20y) and we have
40x + 20y = 120 ... (2)
LINEAR INEQUALITIES
2024-25
Page 2


5 Chapter
vMathematics is the art of saying many things in many
different ways. – MAXWELLv
5.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by translating them in the form of equations. Now a
natural question arises: ‘Is it always possible to translate a statement problem in the
form of an equation? For example, the height of all the students in your class is less
than 160 cm. Your classroom can occupy atmost 60 tables or chairs or both. Here we
get certain statements involving a sign ‘<’ (less than), ‘>’ (greater than), ‘=’ (less than
or equal) and = (greater than or equal) which are known as inequalities.
In this Chapter, we will study linear inequalities in one and two variables. The
study of inequalities is very useful in solving problems in the field of science, mathematics,
statistics, economics, psychology, etc.
5.2 Inequalities
Let us consider the following situations:
(i) Ravi goes to market with ` 200 to buy rice, which is available in packets of 1kg. The
price of one packet of rice is ` 30. If x denotes the number of packets of rice, which he
buys, then the total amount spent by him is ` 30x. Since, he has to buy rice in packets
only, he may not be able to spend the entire amount of ` 200. (Why?) Hence
30x < 200 ... (1)
Clearly the statement (i) is not an equation as it does not involve the sign of equality.
(ii) Reshma has ` 120 and wants to buy some registers and pens. The cost of one
register is  ` 40 and that of a pen is ` 20. In this case, if x denotes the number of
registers and y, the number of pens which Reshma buys, then the total amount spent by
her is ` (40x  +  20y) and we have
40x + 20y = 120 ... (2)
LINEAR INEQUALITIES
2024-25
90       MATHEMATICS
Since in this case the total amount spent may be upto ` 120. Note that the statement (2)
consists of two statements
40x + 20y < 120 ... (3)
and 40x + 20y = 120 ... (4)
Statement (3) is not an equation, i.e., it is an inequality while statement (4) is an equation.
Definition 1 Two real numbers or two algebraic expressions related by the symbol
‘<’, ‘>’, ‘=’ or ‘=’ form an inequality.
Statements such as (1), (2) and (3) above are inequalities.
3 < 5;  7 > 5 are the examples of numerical inequalities while
 x < 5; y > 2; x  =  3,  y =  4 are some examples of  literal inequalities.
3 < 5 < 7 (read as 5 is greater than 3 and less than 7), 3 <  x < 5 (read as x is greater
than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities.
Some more examples of inequalities are:
ax + b < 0 ... (5)
ax + b > 0 ... (6)
ax + b = 0 ... (7)
ax + b = 0 ... (8)
ax + by < c ... (9)
ax + by > c ... (10)
ax + by = c ... (11)
ax + by =  c ... (12)
ax
2
 + bx + c = 0 ... (13)
ax
2
 + bx + c > 0 ... (14)
Inequalities (5), (6), (9), (10) and (14) are strict inequalities while inequalities (7), (8),
(11), (12), and (13) are slack inequalities. Inequalities from (5) to (8) are linear
inequalities in one variable x when a ? 0, while inequalities from (9) to (12) are linear
inequalities in two variables x and y when a ? 0, b ? 0.
Inequalities (13) and (14) are not linear (in fact, these are quadratic inequalities
in one variable x when a ? 0).
In this Chapter, we shall confine ourselves to the study of linear inequalities in one
and two variables only.
2024-25
Page 3


5 Chapter
vMathematics is the art of saying many things in many
different ways. – MAXWELLv
5.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by translating them in the form of equations. Now a
natural question arises: ‘Is it always possible to translate a statement problem in the
form of an equation? For example, the height of all the students in your class is less
than 160 cm. Your classroom can occupy atmost 60 tables or chairs or both. Here we
get certain statements involving a sign ‘<’ (less than), ‘>’ (greater than), ‘=’ (less than
or equal) and = (greater than or equal) which are known as inequalities.
In this Chapter, we will study linear inequalities in one and two variables. The
study of inequalities is very useful in solving problems in the field of science, mathematics,
statistics, economics, psychology, etc.
5.2 Inequalities
Let us consider the following situations:
(i) Ravi goes to market with ` 200 to buy rice, which is available in packets of 1kg. The
price of one packet of rice is ` 30. If x denotes the number of packets of rice, which he
buys, then the total amount spent by him is ` 30x. Since, he has to buy rice in packets
only, he may not be able to spend the entire amount of ` 200. (Why?) Hence
30x < 200 ... (1)
Clearly the statement (i) is not an equation as it does not involve the sign of equality.
(ii) Reshma has ` 120 and wants to buy some registers and pens. The cost of one
register is  ` 40 and that of a pen is ` 20. In this case, if x denotes the number of
registers and y, the number of pens which Reshma buys, then the total amount spent by
her is ` (40x  +  20y) and we have
40x + 20y = 120 ... (2)
LINEAR INEQUALITIES
2024-25
90       MATHEMATICS
Since in this case the total amount spent may be upto ` 120. Note that the statement (2)
consists of two statements
40x + 20y < 120 ... (3)
and 40x + 20y = 120 ... (4)
Statement (3) is not an equation, i.e., it is an inequality while statement (4) is an equation.
Definition 1 Two real numbers or two algebraic expressions related by the symbol
‘<’, ‘>’, ‘=’ or ‘=’ form an inequality.
Statements such as (1), (2) and (3) above are inequalities.
3 < 5;  7 > 5 are the examples of numerical inequalities while
 x < 5; y > 2; x  =  3,  y =  4 are some examples of  literal inequalities.
3 < 5 < 7 (read as 5 is greater than 3 and less than 7), 3 <  x < 5 (read as x is greater
than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities.
Some more examples of inequalities are:
ax + b < 0 ... (5)
ax + b > 0 ... (6)
ax + b = 0 ... (7)
ax + b = 0 ... (8)
ax + by < c ... (9)
ax + by > c ... (10)
ax + by = c ... (11)
ax + by =  c ... (12)
ax
2
 + bx + c = 0 ... (13)
ax
2
 + bx + c > 0 ... (14)
Inequalities (5), (6), (9), (10) and (14) are strict inequalities while inequalities (7), (8),
(11), (12), and (13) are slack inequalities. Inequalities from (5) to (8) are linear
inequalities in one variable x when a ? 0, while inequalities from (9) to (12) are linear
inequalities in two variables x and y when a ? 0, b ? 0.
Inequalities (13) and (14) are not linear (in fact, these are quadratic inequalities
in one variable x when a ? 0).
In this Chapter, we shall confine ourselves to the study of linear inequalities in one
and two variables only.
2024-25
      LINEAR INEQUALITIES       91
5.3 Algebraic Solutions of Linear Inequalities in One Variable and their
Graphical Representation
Let us consider the inequality (1) of Section 6.2, viz, 30x < 200
Note that here x denotes the number of packets of rice.
Obviously, x cannot be a negative integer or a fraction. Left hand side (L.H.S.) of this
inequality is 30x and right hand side (RHS) is 200. Therefore, we have
For x = 0, L.H.S. = 30 (0) = 0 < 200 (R.H.S.), which is true.
For x = 1, L.H.S. = 30 (1) = 30 < 200 (R.H.S.), which is true.
For x = 2, L.H.S. = 30 (2) = 60 < 200, which is true.
For x = 3, L.H.S. = 30 (3) = 90 < 200, which is true.
For x = 4, L.H.S. = 30 (4) = 120 < 200, which is true.
For x = 5, L.H.S. = 30 (5) = 150 < 200, which is true.
For x = 6, L.H.S. = 30 (6) = 180 < 200, which is true.
For x = 7, L.H.S. = 30 (7) = 210 < 200, which is false.
In the above situation, we find that the values of x, which makes the above
inequality a true statement, are 0,1,2,3,4,5,6. These values of x, which make above
inequality a true statement, are called solutions of inequality and the set {0,1,2,3,4,5,6}
is called its solution set.
Thus, any solution of an inequality in one variable is a value of the variable
which makes it a true statement.
We have found the solutions of the above inequality by trial and error method
which is not very efficient. Obviously, this method is time consuming and sometimes
not feasible. W e must have some better or systematic techniques for solving inequalities.
Before that we should go through some more properties of numerical inequalities and
follow them as rules while solving the inequalities.
Y ou will recall that while solving linear equations, we followed the following rules:
Rule 1 Equal numbers may be added to (or subtracted from) both sides of an equation.
Rule 2 Both sides of an equation may be multiplied (or divided) by the same non-zero
number.
In the case of solving inequalities, we again follow the same rules except with a
difference that in Rule 2, the sign of inequality is reversed (i.e., ‘<‘ becomes ‘>’, =’
becomes ‘=’ and so on) whenever we multiply (or divide) both sides of an inequality by
a negative number. It is evident from the facts that
3 > 2 while – 3 < – 2,
– 8 < – 7 while (– 8) (– 2) > (– 7) (– 2) , i.e., 16 > 14.
2024-25
Page 4


5 Chapter
vMathematics is the art of saying many things in many
different ways. – MAXWELLv
5.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by translating them in the form of equations. Now a
natural question arises: ‘Is it always possible to translate a statement problem in the
form of an equation? For example, the height of all the students in your class is less
than 160 cm. Your classroom can occupy atmost 60 tables or chairs or both. Here we
get certain statements involving a sign ‘<’ (less than), ‘>’ (greater than), ‘=’ (less than
or equal) and = (greater than or equal) which are known as inequalities.
In this Chapter, we will study linear inequalities in one and two variables. The
study of inequalities is very useful in solving problems in the field of science, mathematics,
statistics, economics, psychology, etc.
5.2 Inequalities
Let us consider the following situations:
(i) Ravi goes to market with ` 200 to buy rice, which is available in packets of 1kg. The
price of one packet of rice is ` 30. If x denotes the number of packets of rice, which he
buys, then the total amount spent by him is ` 30x. Since, he has to buy rice in packets
only, he may not be able to spend the entire amount of ` 200. (Why?) Hence
30x < 200 ... (1)
Clearly the statement (i) is not an equation as it does not involve the sign of equality.
(ii) Reshma has ` 120 and wants to buy some registers and pens. The cost of one
register is  ` 40 and that of a pen is ` 20. In this case, if x denotes the number of
registers and y, the number of pens which Reshma buys, then the total amount spent by
her is ` (40x  +  20y) and we have
40x + 20y = 120 ... (2)
LINEAR INEQUALITIES
2024-25
90       MATHEMATICS
Since in this case the total amount spent may be upto ` 120. Note that the statement (2)
consists of two statements
40x + 20y < 120 ... (3)
and 40x + 20y = 120 ... (4)
Statement (3) is not an equation, i.e., it is an inequality while statement (4) is an equation.
Definition 1 Two real numbers or two algebraic expressions related by the symbol
‘<’, ‘>’, ‘=’ or ‘=’ form an inequality.
Statements such as (1), (2) and (3) above are inequalities.
3 < 5;  7 > 5 are the examples of numerical inequalities while
 x < 5; y > 2; x  =  3,  y =  4 are some examples of  literal inequalities.
3 < 5 < 7 (read as 5 is greater than 3 and less than 7), 3 <  x < 5 (read as x is greater
than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities.
Some more examples of inequalities are:
ax + b < 0 ... (5)
ax + b > 0 ... (6)
ax + b = 0 ... (7)
ax + b = 0 ... (8)
ax + by < c ... (9)
ax + by > c ... (10)
ax + by = c ... (11)
ax + by =  c ... (12)
ax
2
 + bx + c = 0 ... (13)
ax
2
 + bx + c > 0 ... (14)
Inequalities (5), (6), (9), (10) and (14) are strict inequalities while inequalities (7), (8),
(11), (12), and (13) are slack inequalities. Inequalities from (5) to (8) are linear
inequalities in one variable x when a ? 0, while inequalities from (9) to (12) are linear
inequalities in two variables x and y when a ? 0, b ? 0.
Inequalities (13) and (14) are not linear (in fact, these are quadratic inequalities
in one variable x when a ? 0).
In this Chapter, we shall confine ourselves to the study of linear inequalities in one
and two variables only.
2024-25
      LINEAR INEQUALITIES       91
5.3 Algebraic Solutions of Linear Inequalities in One Variable and their
Graphical Representation
Let us consider the inequality (1) of Section 6.2, viz, 30x < 200
Note that here x denotes the number of packets of rice.
Obviously, x cannot be a negative integer or a fraction. Left hand side (L.H.S.) of this
inequality is 30x and right hand side (RHS) is 200. Therefore, we have
For x = 0, L.H.S. = 30 (0) = 0 < 200 (R.H.S.), which is true.
For x = 1, L.H.S. = 30 (1) = 30 < 200 (R.H.S.), which is true.
For x = 2, L.H.S. = 30 (2) = 60 < 200, which is true.
For x = 3, L.H.S. = 30 (3) = 90 < 200, which is true.
For x = 4, L.H.S. = 30 (4) = 120 < 200, which is true.
For x = 5, L.H.S. = 30 (5) = 150 < 200, which is true.
For x = 6, L.H.S. = 30 (6) = 180 < 200, which is true.
For x = 7, L.H.S. = 30 (7) = 210 < 200, which is false.
In the above situation, we find that the values of x, which makes the above
inequality a true statement, are 0,1,2,3,4,5,6. These values of x, which make above
inequality a true statement, are called solutions of inequality and the set {0,1,2,3,4,5,6}
is called its solution set.
Thus, any solution of an inequality in one variable is a value of the variable
which makes it a true statement.
We have found the solutions of the above inequality by trial and error method
which is not very efficient. Obviously, this method is time consuming and sometimes
not feasible. W e must have some better or systematic techniques for solving inequalities.
Before that we should go through some more properties of numerical inequalities and
follow them as rules while solving the inequalities.
Y ou will recall that while solving linear equations, we followed the following rules:
Rule 1 Equal numbers may be added to (or subtracted from) both sides of an equation.
Rule 2 Both sides of an equation may be multiplied (or divided) by the same non-zero
number.
In the case of solving inequalities, we again follow the same rules except with a
difference that in Rule 2, the sign of inequality is reversed (i.e., ‘<‘ becomes ‘>’, =’
becomes ‘=’ and so on) whenever we multiply (or divide) both sides of an inequality by
a negative number. It is evident from the facts that
3 > 2 while – 3 < – 2,
– 8 < – 7 while (– 8) (– 2) > (– 7) (– 2) , i.e., 16 > 14.
2024-25
92       MATHEMATICS
Thus, we state the following rules for solving an inequality:
Rule 1 Equal numbers may be added to (or subtracted from) both sides of an inequality
without affecting the sign of inequality.
Rule 2  Both sides of an inequality can be multiplied (or divided) by the same positive
number. But when both sides are multiplied or divided by a negative number, then the
sign of inequality is reversed.
Now, let us consider some examples.
Example 1 Solve 30 x < 200 when
(i) x is a natural number, (ii) x is an integer.
Solution We are given  30 x < 200
or
30 200
30 30
x
<
(Rule 2), i.e., x < 20 / 3.
(i) When x is a natural number, in this case the following values of x make the
statement true.
 1, 2, 3, 4, 5, 6.
The solution set of the inequality is {1,2,3,4,5,6}.
(ii) When x is an integer, the solutions of the given inequality are
..., – 3, –2, –1, 0, 1, 2, 3, 4, 5, 6
The solution set of the inequality is {...,–3, –2,–1, 0, 1, 2, 3, 4, 5, 6}
Example 2 Solve 5x – 3 < 3x +1 when
(i) x is an integer, (ii) x is a real number.
Solution We have, 5x –3 < 3x + 1
or 5x –3 + 3 < 3x +1 +3 (Rule 1)
or 5x < 3x +4
or 5x – 3x < 3x + 4 – 3x (Rule 1)
or 2x < 4
or x < 2 (Rule 2)
(i) When x is an integer, the solutions of the given inequality are
..., – 4, – 3, – 2, – 1, 0, 1
(ii) When x is a real number, the solutions of the inequality are given by x < 2,
i.e., all real numbers x which are less than 2. Therefore, the solution set of
the inequality is  x ? (– 8, 2).
We have considered solutions of inequalities in the set of natural numbers, set of
integers and in the set of real numbers. Henceforth, unless stated otherwise, we shall
solve the inequalities in this Chapter in the set of real numbers.
2024-25
Page 5


5 Chapter
vMathematics is the art of saying many things in many
different ways. – MAXWELLv
5.1 Introduction
In earlier classes, we have studied equations in one variable and two variables and also
solved some statement problems by translating them in the form of equations. Now a
natural question arises: ‘Is it always possible to translate a statement problem in the
form of an equation? For example, the height of all the students in your class is less
than 160 cm. Your classroom can occupy atmost 60 tables or chairs or both. Here we
get certain statements involving a sign ‘<’ (less than), ‘>’ (greater than), ‘=’ (less than
or equal) and = (greater than or equal) which are known as inequalities.
In this Chapter, we will study linear inequalities in one and two variables. The
study of inequalities is very useful in solving problems in the field of science, mathematics,
statistics, economics, psychology, etc.
5.2 Inequalities
Let us consider the following situations:
(i) Ravi goes to market with ` 200 to buy rice, which is available in packets of 1kg. The
price of one packet of rice is ` 30. If x denotes the number of packets of rice, which he
buys, then the total amount spent by him is ` 30x. Since, he has to buy rice in packets
only, he may not be able to spend the entire amount of ` 200. (Why?) Hence
30x < 200 ... (1)
Clearly the statement (i) is not an equation as it does not involve the sign of equality.
(ii) Reshma has ` 120 and wants to buy some registers and pens. The cost of one
register is  ` 40 and that of a pen is ` 20. In this case, if x denotes the number of
registers and y, the number of pens which Reshma buys, then the total amount spent by
her is ` (40x  +  20y) and we have
40x + 20y = 120 ... (2)
LINEAR INEQUALITIES
2024-25
90       MATHEMATICS
Since in this case the total amount spent may be upto ` 120. Note that the statement (2)
consists of two statements
40x + 20y < 120 ... (3)
and 40x + 20y = 120 ... (4)
Statement (3) is not an equation, i.e., it is an inequality while statement (4) is an equation.
Definition 1 Two real numbers or two algebraic expressions related by the symbol
‘<’, ‘>’, ‘=’ or ‘=’ form an inequality.
Statements such as (1), (2) and (3) above are inequalities.
3 < 5;  7 > 5 are the examples of numerical inequalities while
 x < 5; y > 2; x  =  3,  y =  4 are some examples of  literal inequalities.
3 < 5 < 7 (read as 5 is greater than 3 and less than 7), 3 <  x < 5 (read as x is greater
than or equal to 3 and less than 5) and 2 < y < 4 are the examples of double inequalities.
Some more examples of inequalities are:
ax + b < 0 ... (5)
ax + b > 0 ... (6)
ax + b = 0 ... (7)
ax + b = 0 ... (8)
ax + by < c ... (9)
ax + by > c ... (10)
ax + by = c ... (11)
ax + by =  c ... (12)
ax
2
 + bx + c = 0 ... (13)
ax
2
 + bx + c > 0 ... (14)
Inequalities (5), (6), (9), (10) and (14) are strict inequalities while inequalities (7), (8),
(11), (12), and (13) are slack inequalities. Inequalities from (5) to (8) are linear
inequalities in one variable x when a ? 0, while inequalities from (9) to (12) are linear
inequalities in two variables x and y when a ? 0, b ? 0.
Inequalities (13) and (14) are not linear (in fact, these are quadratic inequalities
in one variable x when a ? 0).
In this Chapter, we shall confine ourselves to the study of linear inequalities in one
and two variables only.
2024-25
      LINEAR INEQUALITIES       91
5.3 Algebraic Solutions of Linear Inequalities in One Variable and their
Graphical Representation
Let us consider the inequality (1) of Section 6.2, viz, 30x < 200
Note that here x denotes the number of packets of rice.
Obviously, x cannot be a negative integer or a fraction. Left hand side (L.H.S.) of this
inequality is 30x and right hand side (RHS) is 200. Therefore, we have
For x = 0, L.H.S. = 30 (0) = 0 < 200 (R.H.S.), which is true.
For x = 1, L.H.S. = 30 (1) = 30 < 200 (R.H.S.), which is true.
For x = 2, L.H.S. = 30 (2) = 60 < 200, which is true.
For x = 3, L.H.S. = 30 (3) = 90 < 200, which is true.
For x = 4, L.H.S. = 30 (4) = 120 < 200, which is true.
For x = 5, L.H.S. = 30 (5) = 150 < 200, which is true.
For x = 6, L.H.S. = 30 (6) = 180 < 200, which is true.
For x = 7, L.H.S. = 30 (7) = 210 < 200, which is false.
In the above situation, we find that the values of x, which makes the above
inequality a true statement, are 0,1,2,3,4,5,6. These values of x, which make above
inequality a true statement, are called solutions of inequality and the set {0,1,2,3,4,5,6}
is called its solution set.
Thus, any solution of an inequality in one variable is a value of the variable
which makes it a true statement.
We have found the solutions of the above inequality by trial and error method
which is not very efficient. Obviously, this method is time consuming and sometimes
not feasible. W e must have some better or systematic techniques for solving inequalities.
Before that we should go through some more properties of numerical inequalities and
follow them as rules while solving the inequalities.
Y ou will recall that while solving linear equations, we followed the following rules:
Rule 1 Equal numbers may be added to (or subtracted from) both sides of an equation.
Rule 2 Both sides of an equation may be multiplied (or divided) by the same non-zero
number.
In the case of solving inequalities, we again follow the same rules except with a
difference that in Rule 2, the sign of inequality is reversed (i.e., ‘<‘ becomes ‘>’, =’
becomes ‘=’ and so on) whenever we multiply (or divide) both sides of an inequality by
a negative number. It is evident from the facts that
3 > 2 while – 3 < – 2,
– 8 < – 7 while (– 8) (– 2) > (– 7) (– 2) , i.e., 16 > 14.
2024-25
92       MATHEMATICS
Thus, we state the following rules for solving an inequality:
Rule 1 Equal numbers may be added to (or subtracted from) both sides of an inequality
without affecting the sign of inequality.
Rule 2  Both sides of an inequality can be multiplied (or divided) by the same positive
number. But when both sides are multiplied or divided by a negative number, then the
sign of inequality is reversed.
Now, let us consider some examples.
Example 1 Solve 30 x < 200 when
(i) x is a natural number, (ii) x is an integer.
Solution We are given  30 x < 200
or
30 200
30 30
x
<
(Rule 2), i.e., x < 20 / 3.
(i) When x is a natural number, in this case the following values of x make the
statement true.
 1, 2, 3, 4, 5, 6.
The solution set of the inequality is {1,2,3,4,5,6}.
(ii) When x is an integer, the solutions of the given inequality are
..., – 3, –2, –1, 0, 1, 2, 3, 4, 5, 6
The solution set of the inequality is {...,–3, –2,–1, 0, 1, 2, 3, 4, 5, 6}
Example 2 Solve 5x – 3 < 3x +1 when
(i) x is an integer, (ii) x is a real number.
Solution We have, 5x –3 < 3x + 1
or 5x –3 + 3 < 3x +1 +3 (Rule 1)
or 5x < 3x +4
or 5x – 3x < 3x + 4 – 3x (Rule 1)
or 2x < 4
or x < 2 (Rule 2)
(i) When x is an integer, the solutions of the given inequality are
..., – 4, – 3, – 2, – 1, 0, 1
(ii) When x is a real number, the solutions of the inequality are given by x < 2,
i.e., all real numbers x which are less than 2. Therefore, the solution set of
the inequality is  x ? (– 8, 2).
We have considered solutions of inequalities in the set of natural numbers, set of
integers and in the set of real numbers. Henceforth, unless stated otherwise, we shall
solve the inequalities in this Chapter in the set of real numbers.
2024-25
      LINEAR INEQUALITIES       93
Example 3 Solve 4x + 3 < 6x +7.
Solution We have, 4x + 3 < 6x + 7
or 4x – 6x < 6x + 4 – 6x
or – 2x < 4 or   x > – 2
i.e., all the real numbers which are greater than –2, are the solutions of the given
inequality. Hence, the solution set is (–2, 8).
Example 4  Solve 
5 2
5
3 6
– x x
– =
.
Solution We have
      
5 2
5
3 6
– x x
– =
or 2 (5 – 2x)   =  x – 30.
or 10 – 4x = x – 30
or – 5x = – 40,   i.e.,  x  =  8
Thus, all real numbers x which are greater than or equal to 8 are the solutions of the
given inequality, i.e., x ? [8, 8).
Example 5 Solve 7x + 3 < 5x + 9. Show the graph of the solutions on number line.
Solution We have 7x + 3 < 5x + 9 or
 2x < 6 or x < 3
The graphical representation of the solutions are given in Fig 5.1.
Fig 5.1
Example 6   Solve 
3 4 1
1
2 4
x x - +
= -
. Show the graph of the solutions on number line.
Solution We have
3 4 1
1
2 4
x x - +
= -
or
3 4 3
2 4
x x - -
=
or 2 (3x – 4) = (x – 3)
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