JEE Main Important Chemistry formulas
ApplyAs per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters
Access premium articles, webinars, resources to make the best decisions for career, course, exams, scholarships, study abroad and much more with
Plan, Prepare & Make the Best Career Choices
NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities are provided here. These NCERT Solutions are created by expert team at Careers360 keeping in mind of latest syllabus of CBSE 2023-24. 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 linear inequalities class 11 NCERT solutions. Class 11 Mathematics NCERT book will help you understand the concepts in a much easier way. Here you will get NCERT solutions for class 11 also.
JEE Main Scholarship Test Kit (Class 11): Narayana | Physics Wallah | Aakash | Unacademy
Suggested: JEE Main: high scoring chapters | Past 10 year's papers
New: Aakash iACST Scholarship Test. Up to 90% Scholarship. Register Now
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. T he concept of inequality is used in formulating the constraints. In NCERT solutions for class 11 maths chapter 6 linear inequalities you will understand questions based on inequalities in one variable and two variables.
Inequation (Inequality):
An inequation or inequality is a statement involving variables and the sign of inequality like >, <, ≥, or ≤.
Symbols used in inequalities:
The symbol < means less than.
The symbol > means greater than.
The symbol < with a bar underneath ≤ means less than or equal to.
The symbol > with a bar underneath ≥ means greater than or equal to.
The symbol ≠ means the quantities on the left and right sides are not equal to.
Algebraic Solutions for Linear Inequalities in One Variable:
Linear inequalities involve expressions with variables and inequality symbols like <, >, ≤, or ≥.
The solution to a linear inequality can be determined using algebraic methods.
Important rules to follow when solving linear inequalities:
Rule 1: Don’t change the sign of an inequality by adding or subtracting the same integer on both sides of an equation.
Rule 2: Add or subtract the same positive integer from both sides of an inequality equation.
Graphical Representation of Linear Inequalities:
Linear inequalities can also be represented graphically on a number line.
For example, x > 3 represents all real numbers greater than 3, which can be shaded on the number line to the right of 3.
Similarly, x ≤ -2 represents all real numbers less than or equal to -2, which can be shaded on the number line to the left of -2.
Free download NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities for CBSE Exam.
Class 11 maths chapter 6 question answer - Exercise 6.1
Question:1(i) Solve , when
i s a natural number.
Answer:
Given :
Divide by 24 from both sides
i s 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
i s 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
i s 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.
Class 11 maths chapter 6 question answer - 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:
Class 11 maths chapter 6 question answer - Exercise 6.3
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 in cluding 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:
Linear inequalities equations ncert solutions - 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
Definition of Inequality: An inequality is a statement that two values are not equal. In mathematics, inequalities are used to compare values and to represent constraints in real-world problems.
Linear Inequalities: Linear inequalities are a type of inequality where the variables appear only in the first degree, that is, raised to the power of 1.
Solving Linear Inequalities: The process of finding all the possible values of the variable that satisfy a given linear inequality is called solving the inequality. In this chapter, various methods of solving linear inequalities are discussed.
Graphical Representation: Graphical representation is an important tool to visualize the solution of a linear inequality. The chapter 6 class 11 maths discusses how to plot linear inequalities on a coordinate plane.
Solution of System of Linear Inequalities: The NCERT solution for class 11 maths chapter 6 also discusses the solution of a system of linear inequalities, which involves finding the region on the coordinate plane that satisfies all the inequalities in the system.
Application in Real-World Problems: Linear inequalities are widely used in real-world problems, such as optimizing production, minimizing costs, and maximizing profits. The chapter provides various examples of real-world problems that can be solved using linear inequalities.
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
chapter-1 | Sets |
chapter-2 | Relations and Functions |
chapter-3 | Trigonometric Functions |
chapter-4 | Principle of Mathematical Induction |
chapter-5 | Complex Numbers and Quadratic equations |
chapter-6 | Linear Inequalities |
chapter-7 | Permutation and Combinations |
chapter-8 | Binomial Theorem |
chapter-9 | Sequences and Series |
chapter-10 | Straight Lines |
chapter-11 | Conic Section |
chapter-12 | Three Dimensional Geometry |
chapter-13 | Limits and Derivatives |
chapter-14 | Mathematical Reasoning |
chapter-15 | Statistics |
chapter-16 | Probability |
Conceptual Clarity: The chapter begins by introducing the basic concepts of linear inequalities, ensuring that students understand the fundamental principles.
Real-Life Applications: Linear inequalities are explained with reference to real-life scenarios, helping students relate mathematical concepts to practical situations.
Inequality Notations: The chapter covers different types of inequality notations, such as "less than," "greater than," "less than or equal to," and "greater than or equal to".
Happy Reading !!!
Inequalities class 11 includes the important topics such as Basic concept of inequalities, algebraic solutions of linear inequalities in one variable and their graphical representation, graphical solution of linear inequalities in two variables, and solution of system of linear inequalities in two variables. students should practice these concepts to get good a hold of the concepts discussed in class 11 chapter 6.
The steps to plot a graph of a linear inequality covered in linear inequalities class 11 ncert solutions are as follows:
Write the inequality in the form of a linear equation
Solve the equation for y
Identify the boundary line
Choose a test point
Substitute the test point
Shade the region
Identify the solution set
Students can find NCERT solutions for class 11 maths by clicking on the link.
The linear inequalities class 11 solutions includes there three exercises and one miscellaneous exercise.
Exercise 6.1 – 26 Questions
Exercise 6.2 – 10 Questions
Exercise 6.3 – 15 Questions
Miscellaneous Exercise – 14 Questions
NCERT official is the official website of the NCERT where you can get NCERT textbooks and syllabus from class 1 to 12.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.
The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.
How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
An investment director is a person who helps corporations and individuals manage their finances. They can help them develop a strategy to achieve their goals, including paying off debts and investing in the future. In addition, he or she can help individuals make informed decisions.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.
Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities.
Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters
Get up to 90% scholarship on NEET, JEE & Foundation courses
Enroll in Resonance Coaching for success in JEE/NEET exams
Start your JEE preparation with ALLEN
Just Study 32% of the NEET syllabus and Score upto 100% marks
As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE