NCERT solutions for class 11 maths chapter 6 Linear Inequalities: In earlier classes, you have studied equations of one variable and two variables and have solved many problems based on this. In this article, you will get NCERT solutions for class 11 maths chapter 6 linear inequalities. Many real life problems can be solved by converting a problem into a mathematical equation but some problems like the height of all the members in your family is less than 180 cm, auditorium can occupy at most 120 tables or chairs or both can't be converted into equations. Statements which involve sign ‘’ '>' (greater than), ‘≤’ (less than or equal) and ≥ (greater than or equal), '<' (less than) are known as inequalities. The concept of inequality is used in formulating the constraints. In solutions of NCERT for class 11 maths chapter 6 linear inequalities you will understand questions based on inequalities in one variable and two variables. This chapter is useful in various field of mathematics, science and solving real life problems like cost estimation subjected to many constraints. There are 50 problems in 3 exercises of this chapter. All these questions are explained in the CBSE NCERT solutions for class 11 maths chapter 6 linear inequalities in a detailed manner. It will help you understand the concepts in a much easier way. Check all NCERT Solutions from class 6 to 12 which will help you to learn science and maths.
Let's understand this chapter with help of an example.
A manufacturing unit makes two models p and q of a product. Each piece of p requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of q requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The manufacturing unit makes a profit of Rs 8000 on each piece of p and Rs 12000 on each piece of Model q. Formulate this problem in linear equalities to maximize the profit.
The above problem can be formulated using linear inequalities and can be solved using linear programming which you will study in NCERT solutions for class 11 maths chapter 6 linear inequalities.
The above problem is formulated as follows.
Let x is the number of pieces of Model p and y is the number of pieces of Model q
We have to maximize the profit Z= 8000x+12000y subjected to the following constraints
6.1 Introduction
6.2 Inequalities
6.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
6.4 Graphical Solution of Linear Inequalities in Two Variables
6.5 Solution of System of Linear Inequalities in Two Variables
Question:1(i) Solve , when
is a natural number.
Answer:
Given :
Divide by 24 from both sides
is a natural number which is less than 4.167.
Hence, values of x can be
Question:1(ii) Solve , when
Answer:
Given :
Divide by 24 from both sides
is are integers which are less than 4.167.
Hence, values of x can be
Question:2(i) Solve , when
x is a natural number.
Answer:
Given :
Divide by -12 from both side
is a natural number which is less than - 2.5.
Hence, the values of x do not exist for given inequality.
Question:2(ii) Solve , when
Answer:
Given :
Divide by -12 from both side
are integers less than - 2.5 .
Hence, values of x can be
Question:3(i) Solve , when
Answer:
Given :
Divide by 5 from both sides
are integers less than 2
Hence, values of x can be
Question:3(ii) Solve , when
Answer:
Given :
Divide by 5 from both sides
are real numbers less than 2
i.e.
Question:4(i) Solve , when
x is an integer.
Answer:
Given :
Divide by 3 from both sides
are integers greater than -2
Hence, the values of x can be .
Question:4(ii) Solve , when) is a real number.
Answer:
Given :
Divide by 3 from both side
are real numbers greater than -2
Hence , values of x can be as
Question:5 Solve the inequality for real .
Answer:
Given :
are real numbers greater than -4.
Hence, values of x can be as
Question:6 Solve the inequality for real
Answer:
Given :
are real numbers less than -3.
Hence, values of x can be
Question:7 Solve the inequality for real .
Answer:
Given :
are real numbers less than equal to -3
Hence , values of x can be as ,
Question:8 Solve the inequality for real
Answer:
Given :
are real numbers less than equal to 4
Hence, values of x can be as
Question:9 Solve the inequality for real
Answer:
Given :
are real numbers less than 6
Hence, values of x can be as
Question:10 Solve the inequality for real .
Answer:
Given :
are real numbers less than -6
Hence, values of x can be as
Question:11 Solve the inequality for real
Answer:
Given :
are real numbers less than equal to 2.
Hence, values of x can be as
Question:12 Solve the inequality for real
Answer:
Given :
are real numbers less than equal to 120.
Hence, values of x can be as .
Question:13 Solve the inequality for real
Answer:
Given :
are real numbers greater than 4
Hence , values of x can be as
Question:14 Solve the inequality for real
Answer:
Given :
are real numbers less than equal to 2.
Hence , values of x can be as
Question:15 Solve the inequality for real x
Answer:
Given :
are real numbers greater than 4.
Hence, values of x can be as
Question:16 Solve the inequality for real
Answer:
Given :
are real numbers less than equal 2.
Hence, values of x can be as .
Question:17 Solve the inequality and show the graph of the solution on number line
Answer:
Given :
are real numbers less than 3
Hence, values of x can be as
The graphical representation of solutions of the given inequality is as :
Question:18 Solve the inequality and show the graph of the solution on number line
Answer:
Given :
are real numbers greater than equal to -1.
Hence, values of x can be as
The graphical representation of solutions of the given inequality is as :
Question:19 Solve the inequality and show the graph of the solution on number line
Answer:
Given :
are real numbers greater than -1
Hence, values of x can be as
The graphical representation of solutions of given inequality is as :
Question:20 Solve the inequality and show the graph of the solution on number line
Answer:
Given :
are real numbers greater than equal to
Hence, values of x can be as
The graphical representation of solutions of the given inequality is as :
Answer:
Let x be marks obtained by Ravi in the third test.
The student should have an average of at least 60 marks.
the student should have minimum marks of 35 to have an average of 60
Answer:
Sunita’s marks in the first four examinations are 87, 92, 94 and 95.
Let x be marks obtained in the fifth examination.
To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations.
Thus, Sunita must obtain 82 in the fifth examination to get grade ‘A’ in the course.
Answer:
Let x be smaller of two consecutive odd positive integers. Then the other integer is x+2.
Both integers are smaller than 10.
Sum of both integers is more than 11.
We conclude and and x is odd integer number.
x can be 5,7.
The two pairs of consecutive odd positive integers are .
Answer:
Let x be smaller of two consecutive even positive integers. Then the other integer is x+2.
Both integers are larger than 5.
Sum of both integers is less than 23.
We conclude and and x is even integer number.
x can be 6,8,10.
The pairs of consecutive even positive integers are .
Answer:
Let the length of the smallest side be x cm.
Then largest side = 3x cm.
Third side = 3x-2 cm.
Given: The perimeter of the triangle is at least 61 cm.
Minimum length of the shortest side is 9 cm.
Answer:
Let x is the length of the shortest board,
then and are the lengths of the second and third piece, respectively.
The man wants to cut three lengths from a single piece of board of length 91cm.
Thus,
if the third piece is to be at least 5cm longer than the second, than
We conclude that and .
Thus , .
Hence, the length of the shortest board is greater than equal to 8 cm and less than equal to 22 cm.
Solutions of NCERT for class 11 maths chapter 6 linear inequalities-Exercise: 6.2
Question:1 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot in two half planes.
Select a point (not on line ) which lie in one of the half planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane (above the line) is not a solution region of given inequality i.e. .
Also, the point on the line does not satisfy the inequality.
Thus, the solution to this inequality is half plane below the line excluding points on this line represented by the green part.
This can be represented as follows:
Question:2 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot in two half-planes.
Select a point (not on the line ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane II is not a solution region of given inequality i.e.
Also, the point on the line does satisfy the inequality.
Thus, the solution to this inequality is the half plane I, above the line including points on this line , represented by green colour.
This can be represented as follows:
Question:3 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot into two half-planes.
Select a point (not on the line ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, the half plane I(above the line) is not a solution region of given inequality i.e. .
Also, the point on the line does satisfy the inequality.
Thus, the solution to this inequality is half plane II (below the line ) including points on this line, represented by green colour.
This can be represented as follows:
Question:4 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot in two half-planes.
Select a point (not on the line ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane II is not solution region of given inequality i.e. .
Also, the point on the line does satisfy the inequality.
Thus, the solution to this inequality is the half plane I including points on this line, represented by green colour.
This can be represented as follows:
Question:5 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot in two half planes.
Select a point (not on the line ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane Ii is not solution region of given inequality i.e. .
Also, the point on the line does satisfy the inequality.
Thus, the solution to this inequality is the half plane I including points on this line, represented by green colour
This can be represented as follows:
Question:6 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot in two half planes.
Select a point (not on the line )which lie in one of the half-planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is false .
Therefore, half plane I is not solution region of given inequality i.e. .
Also point on line does not satisfy the inequality.
Thus, the solution to this inequality is half plane II excluding points on this line, represented by green colour.
This can be represented as follows:
Question:7 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot in two half planes.
Select a point (not on the line ) which lie in one of the half planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane II is not solution region of given inequality i.e. .
Also, the point on the line does satisfy the inequality.
Thus, the solution to this inequality is the half plane I including points on this line, represented by green colour
This can be represented as follows:
Question:8 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in graph below.
The line divides plot in two half planes.
Select a point (not on the line ) which lie in one of the half plane , to detemine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane II is not solution region of given inequality i.e. .
Also point on the line does not satisfy the inequality.
Thus, solution to this inequality is half plane I excluding points on this line, represented by green colour.
This can be represented as follows:
Question:9 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in graph below.
The line divides plot in two half planes.
Select a point (not on the line ) which lie in one of the half plane , to detemine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is false.
Therefore, the half plane I is not a solution region of given inequality i.e. .
Also, the point on the line does not satisfy the inequality.
Thus, the solution to this inequality is half plane II excluding points on this line, represented by green colour.
This can be represented as follows:
Question:10 Solve the following inequality graphically in two-dimensional plane:
Answer:
Graphical representation of is given in the graph below.
The line divides plot into two half-planes.
Select a point (not on the line ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.
Let there be a point
We observe
i.e. , which is true.
Therefore, half plane II is not a solution region of given inequality i.e. .
Also, the point on the line does not satisfy the inequality.
Thus, the solution to this inequality is the half plane I excluding points on this line.
This can be represented as follows:
Question:1 Solve the following system of inequalities graphically:
Answer:
Graphical representation of and is given in the graph below.
The line and divides plot in four regions i.e.I,II,III,IV.
For ,
The solution to this inequality is region II and III including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region IV and III including points on this line because points on the line also satisfy the inequality.
Hence, solution to is common region of graph i.e. region III.
Thus, solution of is region III.
This can be represented as follows:
The below green colour represents the solution
Question:2 Solve the following system of inequalities graphically:
Answer:
Graphical representation of and is given in graph below.
For ,
The solution to this inequality is region on right hand side of line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on the line also satisfy the inequality.
For
The solution to this inequality is region below the line including points on this line because points on the line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:3 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in the graph below.
For ,
The solution to this inequality is region above line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region below the line including points on this line because points on the line also satisfy the inequality.
Hence, the solution to these linear inequalities is the shaded region(ABC) as shown in figure including points on the respective lines.
This can be represented as follows:
Question:4 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in the graph below.
For ,
The solution to this inequality is region above line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is half plane corresponding to the line containing point excluding points on this line because points on the line does not satisfy the inequality.
Hence, the solution to these linear inequalities is the shaded region as shown in figure including points on line and excluding points on the line.
This can be represented as follows:
Question:5 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in graph below.
For
The solution to this inequality is region below line excluding points on this line because points on line does not satisfy the inequality.
For ,
The solution to this inequality is region above the line excluding points on this line because points on line does not satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure excluding points on the lines.
This can be represented as follows:
Question:6 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in the graph below.
For
The solution to this inequality is region below line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on the line also satisfy the inequality.
Hence, the solution to these linear inequalities is shaded region as shown in figure including points on the lines.
This can be represented as follows:
Question:7 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in graph below.
For
The solution to this inequality is region above line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the lines.
This can be represented as follows:
Question:8 Solve the following system of inequalities graphically:
Answer:
Graphical representation of and is given in graph below.
For ,
The solution to this inequality is region below line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality represents half plane corresponding to the line containing point excluding points on this line because points on line does not satisfy the inequality.
For ,
The solution to this inequality is region on right hand side of the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure.
This can be represented as follows:
Question:9 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in graph below.
For ,
The solution to this inequality is region below the line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region right hand side of the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:10 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in graph below.
For ,
The solution to this inequality is region below the line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region below the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region right hand side of the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, the solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:11 Solve the following system of inequalities graphically:
Answer:
Graphical representation of and is given in graph below.
For ,
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region below the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:12 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in graph below.
For ,
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region right hand side of the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:13 Solve the following system of inequalities graphically:
Answer:
Graphical representation of is given in graph below.
For
The solution to this inequality is region below the line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region right hand side of the line including points on this line because points on the line also satisfy the inequality.
For
The solution to this inequality is region right hand side of the line including points on this line because points on the line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:14 Solve the following system of inequality graphically:
Answer:
Graphical representation of is given in graph below.
For
The solution to this inequality is region below the line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region below the line including points on this line because points on the line also satisfy the inequality.
For ,
The solution to this inequality is region left hand side of the line including points on this line because points on the line also satisfy the inequality.
For
The solution to this inequality is region right hand side of the line including points on this line because points on the line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
Question:15 Solve the following system of inequality graphically:
Answer:
Graphical representation of is given in graph below.
For
The solution to this inequality is region below the line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
For ,
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region right hand side of the line including points on this line because points on line also satisfy the inequality.
For
The solution to this inequality is region above the line including points on this line because points on line also satisfy the inequality.
Hence, solution to these linear inequalities is shaded region as shown in figure including points on the respective lines.
This can be represented as follows:
NCERT solutions for class 11 maths chapter 6 linear inequalities-Miscellaneous Exercise
Question:1 Solve the inequality
Answer:
Given :
Thus, all the real numbers greater than equal to 2 and less than equal to 3 are solutions to this inequality.
Solution set is
Question:6 Solve the inequality
Answer:
Given the linear inequality
The solution set of the given inequality is
Question:7 Solve the inequality and represent the solution graphically on number line.
Answer:
Given :
The solution graphically on the number line is as shown :
Question:8 Solve the inequality and represent the solution graphically on number line.
Answer:
Given :
The solution graphically on the number line is as shown :
Question:9 Solve the inequality and represent the solution graphically on number line.
Answer:
Given :
The solution graphically on the number line is as shown :
Question:10 Solve the inequality and represent the solution graphically on number line.
Answer:
Given :
The solution graphically on the number line is as shown :
Answer:
Since the solution is to be kept between 68° F and 77° F.
Putting the value of , we have
the range in temperature in degree Celsius (C) is between 20 to 25.
Answer:
Let x litres of 2% boric acid solution is required to be added.
Total mixture = (x+640) litres
The resulting mixture is to be more than 4% but less than 6% boric acid.
and
and
Thus, the number of litres 2% of boric acid solution that is to be added will have to be more than 320 and less than 1280 litres.
Answer:
Let x litres of water is required to be added.
Total mixture = (x+1125) litres
It is evident that amount of acid contained in the resulting mixture is 45% of 1125 litres.
The resulting mixture contain more than 25 % but less than 30% acid.
and
and
Thus, the number of litres water that is to be added will have to be more than 562.5 and less than 900 litres.
Answer:
Given that group of 12 years old children.
For a group of 12 years old children, CA =12 years
Putting the value of IQ, we obtain
Thus, the range of mental age of the group of 12 years old children is
NCERT solutions for class 11 maths chapter 6 linear inequalities will build your fundamentals which will be helpful in solving many real-life problems like maximizing the profit, minimizing the expenditure, allocating the resources with given constraints.
As all the above questions are prepared and explained in a step-by-step manner with the help of the graphs, it can be understood and visualize the problem easily.
CBSE NCERT solutions for class 11 maths chapter 6 linear inequalities will some innovative ways of solving the problems which become very important to solve some specific problems in an easy way.
This chapter also useful in the prediction of future events based on the past data which is the fundamentals of machine learning
Happy Reading !!!