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    NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities

    Edited By Ramraj Saini | Updated on Sep 23, 2023 06:01 PM IST

    Linear Inequalities Class 11 Questions And Answers

    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.

    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.

    Linear Inequalities Class 11 Questions And Answers PDF Free Download

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    Linear Inequalities Class 11 Solutions - Important Formulae And Points

    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.

    Linear Inequalities Class 11 NCERT Solutions (Intext Questions and Exercise)

    Class 11 maths chapter 6 question answer - Exercise 6.1

    Question:1(i) Solve 24x < 100 , when
    x i s a natural number.

    Answer:

    Given : 24x < 100

    \Rightarrow 24x < 100

    Divide by 24 from both sides

    \Rightarrow \, \, \, \frac{24}{24}x< \frac{100}{24}

    \Rightarrow \, \, \, x< \frac{25}{6}

    \Rightarrow \, \, \, x< 4.167

    x i s a natural number which is less than 4.167.

    Hence, values of x can be \left \{ 1,2,3,4 \right \}

    Question:1(ii) Solve 24x< 100 , when

    x is an integer.

    Answer:

    Given : 24x < 100

    \Rightarrow 24x < 100

    Divide by 24 from both sides

    \Rightarrow \, \, \, \frac{24}{24}x< \frac{100}{24}

    \Rightarrow \, \, \, x< \frac{25}{6}

    \Rightarrow \, \, \, x< 4.167


    x i s are integers which are less than 4.167.

    Hence, values of x can be \left \{..........-3,-2,-1,0, 1,2,3,4 \right \}

    Question:2(i) Solve -12x>30 , when
    x is a natural number.

    Answer:

    Given : -12x>30

    \Rightarrow -12x>30

    Divide by -12 from both side

    \Rightarrow \, \, \, \frac{-12}{-12}x< \frac{30}{-12}

    \Rightarrow \, \, \, x< \frac{30}{-12}

    \Rightarrow \, \, \, x< -2.5

    x 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 - 12x > 30 , when

    x is an integer.

    Answer:

    Given : -12x>30

    \Rightarrow -12x>30

    Divide by -12 from both side

    \Rightarrow \, \, \, \frac{-12}{-12}x< \frac{30}{-12}

    \Rightarrow \, \, \, x< \frac{30}{-12}

    \Rightarrow \, \, \, x< -2.5

    x are integers less than - 2.5 .

    Hence, values of x can be \left \{ .............,-6,-5,-4,-3 \right \}

    Question:3(i) Solve 5x - 3 < 7 , when

    x is an integer.

    Answer:

    Given : 5x - 3 < 7

    \Rightarrow 5x - 3 < 7

    \Rightarrow \, \, \, 5x< 10

    Divide by 5 from both sides

    \Rightarrow \, \, \, \frac{5}{5}x< \frac{10}{5}

    \Rightarrow \, \, \, x< 2

    x are integers less than 2

    Hence, values of x can be \left \{.........-3,-2-1,0,1,\right \}

    Question:3(ii) Solve 5x - 3 < 7 , when

    x is a real number.

    Answer:

    Given : 5x - 3 < 7

    \Rightarrow 5x - 3 < 7

    \Rightarrow \, \, \, 5x< 10

    Divide by 5 from both sides

    \Rightarrow \, \, \, \frac{5}{5}x< \frac{10}{5}

    \Rightarrow \, \, \, x< 2

    x are real numbers less than 2

    i.e. x\in (-\infty ,2)

    Question:4(i) Solve 3x + 8 >2 , when
    x is an integer.

    Answer:

    Given : 3x + 8 >2

    \Rightarrow 3x + 8 >2

    \Rightarrow \, \, \, 3x> -6

    Divide by 3 from both sides

    \Rightarrow \, \, \, \frac{3}{3}x> \frac{-6}{3}

    \Rightarrow \, \, \, x> - 2


    x are integers greater than -2

    Hence, the values of x can be \left \{-1,0,1,2,3,4...............\right \} .

    Question:4(ii) Solve 3x + 8 >2 , when ) x is a real number.

    Answer:

    Given : 3x + 8 >2

    \Rightarrow 3x + 8 >2

    \Rightarrow \, \, \, 3x> -6

    Divide by 3 from both side

    \Rightarrow \, \, \, \frac{3}{3}x> \frac{-6}{3}

    \Rightarrow \, \, \, x> - 2


    x are real numbers greater than -2

    Hence , values of x can be as x\in (-2,\infty )

    Question:5 Solve the inequality for real x . 4x + 3 < 5x + 7

    Answer:

    Given : 4x + 3 < 5x + 7

    \Rightarrow4x + 3 < 5x + 7

    \Rightarrow \, \, \, 4x-5x< 7-3

    \Rightarrow \, \, \, x> -4

    x are real numbers greater than -4.

    Hence, values of x can be as x\in (-4 ,\infty )

    Question:6 Solve the inequality for real x 3x - 7 > 5x -1

    Answer:

    Given : 3x - 7 > 5x -1

    \Rightarrow3x - 7 > 5x -1

    \Rightarrow \, \, \, -2x> 6

    \Rightarrow \, \, \, x< \frac{6}{-2}

    \Rightarrow \, \, \, x< -3

    x are real numbers less than -3.

    Hence, values of x can be x\in (-\infty ,-3)

    Question:7 Solve the inequality for real x . 3(x-1) \leq 2(x-3)

    Answer:

    Given : 3(x-1) \leq 2(x-3)

    \Rightarrow3(x-1) \leq 2(x-3)

    \Rightarrow \, \, \, 3x-3\leq 2x-6

    \Rightarrow \, \, \, 3x-2x\leq -6+3

    \Rightarrow \, \, \, x\leq -3

    x are real numbers less than equal to -3

    Hence , values of x can be as , x\in (-\infty ,-3]

    Question:8 Solve the inequality for real x 3(2- x) \geq 2(1-x)

    Answer:

    Given : 3(2- x) \geq 2(1-x)

    \Rightarrow3(2- x) \geq 2(1-x)

    \Rightarrow \, \, \, 6-3x\geq 2-2x

    \Rightarrow \, \, \, 6-2\geq 3x-2x

    \Rightarrow \, \, \, 4\geq x

    x are real numbers less than equal to 4

    Hence, values of x can be as x\in (-\infty ,4]

    Question:9 Solve the inequality for real x x + \frac{x}{2} + \frac{x}{3} < 11

    Answer:

    Given : x + \frac{x}{2} + \frac{x}{3} < 11

    \Rightarrowx + \frac{x}{2} + \frac{x}{3} < 11

    \Rightarrow \, \, \, x(1+\frac{1}{2}+\frac{1}{3})< 11

    \Rightarrow \, \, \, x(\frac{11}{6})< 11

    \Rightarrow \, \, \, 11 x< 11\times 6

    \Rightarrow \, \, \, x< 6

    x are real numbers less than 6

    Hence, values of x can be as x\in (-\infty ,6)

    Question:10 Solve the inequality for real x . \frac{x}{3} > \frac{x}{2} + 1

    Answer:

    Given : \frac{x}{3} > \frac{x}{2} + 1

    \Rightarrow\frac{x}{3} > \frac{x}{2} + 1

    \Rightarrow \, \, \, \frac{x}{3}-\frac{x}{2}> 1

    \Rightarrow \, \, \,x (\frac{1}{3}-\frac{1}{2})> 1

    \Rightarrow \, \, \,x (-\frac{1}{6})> 1

    \Rightarrow \, \, \, -x > 6

    \Rightarrow \, \, \, x< -6

    x are real numbers less than -6

    Hence, values of x can be as x\in (-\infty ,-6)

    Question:11 Solve the inequality for real x \frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}

    Answer:

    Given : \frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}

    \Rightarrow\frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}

    \Rightarrow \, \, \, 9(x-2)\leq 25(2-x)

    \Rightarrow \, \, \, 9x-18\leq 50-25x

    \Rightarrow \, \, \, 9x+25x\leq 50+18

    \Rightarrow \, \, \, 34x\leq 68

    \Rightarrow \, \, \, x\leq 2

    x are real numbers less than equal to 2.

    Hence, values of x can be as x\in (-\infty ,2]

    Question:12 Solve the inequality for real x \frac{1}{2}\left(\frac{3x}{5} + 4 \right ) \geq \frac{1}{3}(x - 6)

    Answer:

    Given : \frac{1}{2}\left(\frac{3x}{5} + 4 \right ) \geq \frac{1}{3}(x - 6)

    \Rightarrow\frac{1}{2}\left(\frac{3x}{5} + 4 \right ) \geq \frac{1}{3}(x - 6)

    \Rightarrow \, \, 3\left(\frac{3x}{5} + 4 \right ) \geq 2(x - 6)

    \Rightarrow \, \, \frac{9x}{5} + 12 \geq 2x-12

    \Rightarrow \, \, 12+12 \geq 2x-\frac{9x}{5}

    \Rightarrow \, \, 24 \geq \frac{x}{5}

    \Rightarrow \, \, 120 \geq x

    x are real numbers less than equal to 120.

    Hence, values of x can be as x\in (-\infty,120 ] .

    Question:13 Solve the inequality for real x 2(2x + 3) - 10 < 6(x-2)

    Answer:

    Given : 2(2x + 3) - 10 < 6(x-2)

    \Rightarrow2(2x + 3) - 10 < 6(x-2)

    \Rightarrow \, \, \, 4x+6-10 < 6x-12

    \Rightarrow \, \, \, 6-10+12 < 6x-4x

    \Rightarrow \, \, \, 8 < 2x

    \Rightarrow \, \, \, 4 < x

    x are real numbers greater than 4

    Hence , values of x can be as x\in (4,\infty )

    Question:14 Solve the inequality for real x 37 - (3x + 5) \geq 9x - 8(x-3)

    Answer:

    Given : 37 - (3x + 5) \geq 9x - 8(x-3)

    \Rightarrow37 - (3x + 5) \geq 9x - 8(x-3)

    \Rightarrow \, \, \, 37 - 3x - 5 \geq 9x - 8x+24

    \Rightarrow \, \, \, 32 - 3x \geq x+24

    \Rightarrow \, \, \, 32 - 24 \geq x+3x

    \Rightarrow \, \, \, 8 \geq 4x

    \Rightarrow \, \, \, 2\geq x

    x are real numbers less than equal to 2.

    Hence , values of x can be as x\in (-\infty ,2]

    Question:15 Solve the inequality for real x \frac{x}{4}< \frac{(5x-2)}{3} - \frac{(7x-3)}{5}

    Answer:

    Given : \frac{x}{4}< \frac{(5x-2)}{3} - \frac{(7x-3)}{5}

    \Rightarrow \frac{x}{4}< \frac{(5x-2)}{3} - \frac{(7x-3)}{5}

    \Rightarrow \, \, \, \, 15x< 20(5x-2)-12(7x-3)

    \Rightarrow \, \, \, \, 15x< 100x-40-84x+36

    \Rightarrow \, \, \, \, 15x< 16x-4

    \Rightarrow \, \, \, \, 4< x

    x are real numbers greater than 4.

    Hence, values of x can be as x\in (4,\infty)

    Question:16 Solve the inequality for real x \frac{(2x - 1)}{3} \geq \frac{3x-2}{4} - \frac{(2-x)}{5}

    Answer:

    Given : \frac{(2x - 1)}{3} \geq \frac{3x-2}{4} - \frac{(2-x)}{5}

    \Rightarrow\frac{(2x - 1)}{3} \geq \frac{3x-2}{4} - \frac{(2-x)}{5}

    \Rightarrow \, \, \, 20(2x - 1) \geq 15(3x-2) - 12(2-x)

    \Rightarrow \, \, \, 40x - 20 \geq 45x-30 - 24+12x

    \Rightarrow \, \, \, 30+24 - 20 \geq 45x-40x+12x

    \Rightarrow \, \, \, 34 \geq 17x

    \Rightarrow \, \, \, 2 \geq x

    x are real numbers less than equal 2.

    Hence, values of x can be as x\in (-\infty,2 ] .

    Question:17 Solve the inequality and show the graph of the solution on number line 3x - 2 < 2x + 1

    Answer:

    Given : 3x - 2 < 2x + 1

    \Rightarrow3x - 2 < 2x + 1

    \Rightarrow \, \, \, 3x - 2x< 2 + 1

    \Rightarrow \, \, \, x< 3

    x are real numbers less than 3

    Hence, values of x can be as x\in (-\infty ,3)

    The graphical representation of solutions of the given inequality is as :

    1635762263704

    Question:18 Solve the inequality and show the graph of the solution on number line 5x - 3 \geq 3x -5

    Answer:

    Given : 5x - 3 \geq 3x -5

    \Rightarrow5x - 3 \geq 3x -5

    \Rightarrow \, \, \, 5x - 3x \geq 3 -5

    \Rightarrow \, \, \, 2x \geq -2

    \Rightarrow \, \, \, x \geq -1

    x are real numbers greater than equal to -1.

    Hence, values of x can be as x\in [-1,\infty )

    The graphical representation of solutions of the given inequality is as :

    1635762285990

    Question:19 Solve the inequality and show the graph of the solution on number line 3(1-x) < 2 (x +4)

    Answer:

    Given : 3(1-x) < 2 (x +4)

    \Rightarrow3(1-x) < 2 (x +4)

    \Rightarrow \, \, \, 3- 3x< 2x + 8

    \Rightarrow \, \, \, 3- 8< 2x + 3x

    \Rightarrow \, \, \, -5< 5 x

    \Rightarrow \, \, \, -1< x

    x are real numbers greater than -1

    Hence, values of x can be as x\in (-1,\infty )

    The graphical representation of solutions of given inequality is as :

    1635762322180

    Question:20 Solve the inequality and show the graph of the solution on number line \frac{x}{2} \geq \frac{(5x-2)}{3} - \frac{(7x-3)}{5}

    Answer:

    Given : \frac{x}{2} \geq \frac{(5x-2)}{3} - \frac{(7x-3)}{5}

    \Rightarrow\frac{x}{2} \geq \frac{(5x-2)}{3} - \frac{(7x-3)}{5}

    \Rightarrow \, \, \, 15x \geq 10(5x-2) - 6(7x-3)

    \Rightarrow \, \, \, 15x \geq 50x-20 - 42x+18

    \Rightarrow \, \, \, 15x+42x-50x \geq 18-20

    \Rightarrow \, \, \, 7x \geq -2

    \Rightarrow \, \, \, x \geq \frac{-2}{7}

    x are real numbers greater than equal to = \frac{-2}{7}

    Hence, values of x can be as x\in (-\frac{2}{7},\infty )

    The graphical representation of solutions of the given inequality is as :

    1635762357828

    Question:21 Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

    Answer:

    Let x be marks obtained by Ravi in the third test.

    The student should have an average of at least 60 marks.

    \therefore \, \, \, \frac{70+75+x}{3}\geq 60

    \, \, \, 145+x\geq 180

    x\geq 180-145

    x\geq 35

    the student should have minimum marks of 35 to have an average of 60

    Question:22 To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the course.

    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.

    \therefore \, \, \, \frac{87+92+94+95+x}{5}\geq 90

    \Rightarrow \, \, \, \frac{368+x}{5}\geq 90

    \Rightarrow \, \, \, 368+x\geq 450

    \Rightarrow \, \, \, x\geq 450-368

    \Rightarrow \, \, \, x\geq 82

    Thus, Sunita must obtain 82 in the fifth examination to get grade ‘A’ in the course.

    Question:23 Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11.

    Answer:

    Let x be smaller of two consecutive odd positive integers. Then the other integer is x+2.

    Both integers are smaller than 10.

    \therefore \, \, \, x+2< 10

    \Rightarrow \, \, \, \, x< 10-2

    \Rightarrow \, \, \, \, x< 8

    Sum of both integers is more than 11.

    \therefore \, \, \, x+(x+2)> 11

    \Rightarrow \, \, \, (2x+2)> 11

    \Rightarrow \, \, \, 2x> 11-2

    \Rightarrow \, \, \, 2x> 9

    \Rightarrow \, \, \, x> \frac{9}{2}

    \Rightarrow \, \, \, x> 4.5

    We conclude \, \, \, \, x< 8 and \, \, \, x> 4.5 and x is odd integer number.

    x can be 5,7.

    The two pairs of consecutive odd positive integers are (5,7)\, \, \, and\, \, \, (7,9) .

    Question:24 Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

    Answer:

    Let x be smaller of two consecutive even positive integers. Then the other integer is x+2.

    Both integers are larger than 5.

    \therefore \, \, \, x> 5

    Sum of both integers is less than 23.

    \therefore \, \, \, x+(x+2)< 23

    \Rightarrow \, \, \, (2x+2)< 23

    \Rightarrow \, \, \, 2x< 23-2

    \Rightarrow \, \, \, 2x< 21

    \Rightarrow \, \, \, x< \frac{21}{2}

    \Rightarrow \, \, \, x< 10.5

    We conclude \, \, \, \, x< 10.5 and \, \, \, x> 5 and x is even integer number.

    x can be 6,8,10.

    The pairs of consecutive even positive integers are (6,8),(8,10),(10,12) .

    Question:25 The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.

    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.

    \therefore\, \, \, x+3x+(3x-2)\geq 61

    \Rightarrow \, \, \, 7x-2\geq 61

    \Rightarrow \, \, \, 7x\geq 61+2

    \Rightarrow \, \, \, 7x\geq 63

    \Rightarrow \, \, \, x\geq \frac{63}{7}

    \Rightarrow \, \, \, x\geq 9

    Minimum length of the shortest side is 9 cm.

    Question:26 A man wants to cut three lengths from a single piece of board of length 91cm. The second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5cm longer than the second?

    [ Hint : If x is the length of the shortest board, then x , (x + 3) and 2x are the lengths of the second and third piece, respectively. Thus, x + (x + 3) + 2x \leq 91 and 2x \geq (x + 3) + 5 ].

    Answer:

    Let x is the length of the shortest board,

    then (x + 3) and 2x 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, x + (x + 3) + 2x \leq 91

    4x+3\leq 91

    \Rightarrow \, \, \, \, 4x\leq 91-3

    \Rightarrow \, \, \, \, 4x\leq 88

    \Rightarrow \, \, \, \, x\leq \frac{88}{4}

    \Rightarrow \, \, \, \, x\leq 22

    if the third piece is to be at least 5cm longer than the second, than

    2x \geq (x + 3) + 5

    \Rightarrow \, \, \, \, 2x\geq x+8

    \Rightarrow \, \, \, \, 2x-x\geq 8

    \Rightarrow \, \, \, \, x\geq 8

    We conclude that \, \, \, \, x\geq 8 and \, \, \, \, x\leq 22 .

    Thus , 8\leq x\leq 22 .

    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:

    x + y < 5

    Answer:

    Graphical representation of x+y=5 is given in the graph below.

    The line x+y=5 divides plot in two half planes.

    Select a point (not on line x+y=5 ) which lie in one of the half planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    1+2< 5 i.e. 3< 5 , which is true.

    Therefore, half plane (above the line) is not a solution region of given inequality i.e. x + y < 5 .

    Also, the point on the line does not satisfy the inequality.

    Thus, the solution to this inequality is half plane below the line x+y=5 excluding points on this line represented by the green part.

    This can be represented as follows:

    1635762532191

    Question:2 Solve the following inequality graphically in two-dimensional plane: 2x + y \geq 6

    Answer:

    2x + y \geq 6

    Graphical representation of 2x+y=6 is given in the graph below.

    The line 2x+y=6 divides plot in two half-planes.

    Select a point (not on the line 2x+y=6 ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.

    Let there be a point (3,2)

    We observe

    6+2\geq 6 i.e. 8\geq 6 , which is true.

    Therefore, half plane II is not a solution region of given inequality i.e. 2x + y \geq 6

    Also, the point on the line does satisfy the inequality.

    Thus, the solution to this inequality is the half plane I, above the line 2x+y=6 including points on this line , represented by green colour.

    This can be represented as follows:

    1635762558839

    Question:3 Solve the following inequality graphically in two-dimensional plane: 3x + 4y \leq 12

    Answer:

    3x + 4y \leq 12

    Graphical representation of 3x + 4y = 12 is given in the graph below.

    The line 3x + 4y = 12 divides plot into two half-planes.

    Select a point (not on the line 3x + 4y = 12 ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    1+2\leq 12 i.e. 3\leq 12 , which is true.

    Therefore, the half plane I(above the line) is not a solution region of given inequality i.e. 3x + 4y \leq 12 .

    Also, the point on the line does satisfy the inequality.

    Thus, the solution to this inequality is half plane II (below the line 3x + 4y = 12 ) including points on this line, represented by green colour.

    This can be represented as follows:

    1635762572337

    Question:4 Solve the following inequality graphically in two-dimensional plane: y + 8 \geq 2x

    Answer:

    y + 8 \geq 2x

    Graphical representation of y + 8 = 2x is given in the graph below.

    The line y + 8 = 2x divides plot in two half-planes.

    Select a point (not on the line y + 8 = 2x ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    2+8\geq 2\times 1 i.e. 10\geq 2 , which is true.

    Therefore, half plane II is not solution region of given inequality i.e. y + 8 \geq 2x .

    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:

    1635762587304

    Question:5 Solve the following inequality graphically in two-dimensional plane: x - y \leq 2

    Answer:

    x - y \leq 2

    Graphical representation of x - y =2 is given in the graph below.

    The line x - y =2 divides plot in two half planes.

    Select a point (not on the line x - y =2 ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    1-2\leq 2 i.e. -1\leq 2 , which is true.

    Therefore, half plane Ii is not solution region of given inequality i.e. x - y \leq 2 .

    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:

    1635762601555

    Question:6 Solve the following inequality graphically in two-dimensional plane: 2x - 3y > 6

    Answer:

    2x - 3y > 6

    Graphical representation of 2x - 3y = 6 is given in the graph below.

    The line 2x - 3y = 6 divides plot in two half planes.

    Select a point (not on the line 2x - 3y = 6 )which lie in one of the half-planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    2-6> 6 i.e. -4 > 6 , which is false .

    Therefore, half plane I is not solution region of given inequality i.e. 2x - 3y > 6 .

    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:

    1635762622339

    Question:7 Solve the following inequality graphically in two-dimensional plane: -3x + 2y \geq -6

    Answer:

    -3x + 2y \geq -6

    Graphical representation of -3x + 2y = -6 is given in the graph below.

    The line -3x + 2y = -6 divides plot in two half planes.

    Select a point (not on the line -3x + 2y = -6 ) which lie in one of the half planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    -3+4\geq -6 i.e. 1\geq -6 , which is true.

    Therefore, half plane II is not solution region of given inequality i.e. -3x + 2y \geq -6 .

    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:

    1635762636448

    Question:8 Solve the following inequality graphically in two-dimensional plane: 3y - 5x < 30

    Answer:

    3y - 5x < 30

    Graphical representation of 3y - 5x =30 is given in graph below.

    The line 3y - 5x =30 divides plot in two half planes.

    Select a point (not on the line 3y - 5x =30 ) which lie in one of the half plane , to detemine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    6-5< 30 i.e. 1< 30 , which is true.

    Therefore, half plane II is not solution region of given inequality i.e. 3y - 5x < 30 .

    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:

    1635762649943

    Question:9 Solve the following inequality graphically in two-dimensional plane: y < -2

    Answer:

    y < -2

    Graphical representation of y=-2 is given in graph below.

    The line y < -2 divides plot in two half planes.

    Select a point (not on the line y < -2 ) which lie in one of the half plane , to detemine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    i.e. 2< -2 , which is false.

    Therefore, the half plane I is not a solution region of given inequality i.e. y < -2 .

    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:

    1635762675271

    Question:10 Solve the following inequality graphically in two-dimensional plane: x > - 3

    Answer:

    x > - 3

    Graphical representation of x=-3 is given in the graph below.

    The line x=-3 divides plot into two half-planes.

    Select a point (not on the line x=-3 ) which lie in one of the half-planes, to determine whether the point satisfies the inequality.

    Let there be a point (1,2)

    We observe

    i.e. 1> -3 , which is true.

    Therefore, half plane II is not a solution region of given inequality i.e. x > - 3 .

    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:

    1635762691903


    Class 11 maths chapter 6 question answer - Exercise 6.3

    Question:1 Solve the following system of inequalities graphically:

    x \geq 3,\ y\geq 2

    Answer:

    x \geq 3,\ y\geq 2

    Graphical representation of x=3 and y=2 is given in the graph below.

    The line x=3 and y=2 divides plot in four regions i.e.I,II,III,IV.

    For x \geq 3 ,

    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 y \geq 2 ,

    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 x \geq 3,\ y\geq 2 is common region of graph i.e. region III.

    Thus, solution of x \geq 3,\ y\geq 2 is region III.

    This can be represented as follows:

    1654687280859

    The below green colour represents the solution

    1654687281313

    Question:2 Solve the following system of inequalities graphically: 3x +2y \leq 12,\ x \geq 1, \ y\geq 2

    Answer:

    3x +2y \leq 12,\ x \geq 1, \ y\geq 2

    Graphical representation of x=1 \, \, ,3x+2y=12 and y=2 is given in graph below.

    For x \geq 1 ,

    The solution to this inequality is region on right hand side of line (x=1) including points on this line because points on the line also satisfy the inequality.

    For y \geq 2 ,

    The solution to this inequality is region above the line (y=2) including points on this line because points on the line also satisfy the inequality.

    For 3x+2y\leq 12

    The solution to this inequality is region below the line (3x+2y= 12) 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:

    1635762856044

    Question:3 Solve the following system of inequalities graphically: 2x +y \geq 6, 3x +4y\leq 12

    Answer:

    2x +y \geq 6, 3x +4y\leq 12

    Graphical representation of 2x +y =6\, \, and\, \, 3x +4y=12 is given in the graph below.

    For 2x +y \geq 6 ,

    The solution to this inequality is region above line (2x +y =6) including points on this line because points on the line also satisfy the inequality.

    For 3x +4y\leq 12 ,

    The solution to this inequality is region below the line ( 3x +4y= 12) 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:

    1635762935162

    Question:4 Solve the following system of inequalities graphically: x + y \geq 4, 2x - y <0

    Answer:

    x + y \geq 4, 2x - y <0

    Graphical representation of x +y =4\, \, and\, \, 2x -y=0 is given in the graph below.

    For x + y \geq 4, ,

    The solution to this inequality is region above line (x +y =4) including points on this line because points on the line also satisfy the inequality.

    For 2x - y <0 ,

    The solution to this inequality is half plane corresponding to the line ( 2x -y=0) containing point (1,0) 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 (x +y =4) and excluding points on the line ( 2x -y=0) .

    This can be represented as follows:

    1635763100696

    Question:5 Solve the following system of inequalities graphically: 2x - y > 1, \ x -2y < -1

    Answer:

    2x - y > 1, \ x -2y < -1

    Graphical representation of x -2y =-1\, \, and\, \, 2x -y=1 is given in graph below.

    For 2x - y > 1,

    The solution to this inequality is region below line ( 2x -y=1) excluding points on this line because points on line does not satisfy the inequality.

    For \ x -2y < -1 ,

    The solution to this inequality is region above the line (x -2y =-1) 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:

    cz1635763130508

    Question:6 Solve the following system of inequalities graphically: x + y \leq 6, x + y \geq 4

    Answer:

    x + y \leq 6, x + y \geq 4

    Graphical representation of x + y = 6,\, \, and\, \, \, x + y = 4 is given in the graph below.

    For x + y \leq 6,

    The solution to this inequality is region below line ( x+y=6) in cluding points on this line because points on the line also satisfy the inequality.

    For x + y \geq 4 ,

    The solution to this inequality is region above the line ( x+y=4) 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:

    1635763165801

    Question:7 Solve the following system of inequalities graphically: 2x + y \geq 8 , x + 2y \geq 10

    Answer:

    2x + y \geq 8 , x + 2y \geq 10

    Graphical representation of 2x + y = 8\, \, and\, \, x + 2y =10 is given in graph below.

    For 2x + y \geq 8 ,

    The solution to this inequality is region above line (2x + y = 8) including points on this line because points on line also satisfy the inequality.

    For x + 2y \geq 10 ,

    The solution to this inequality is region above the line ( x + 2y =10) 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:

    1635763179656

    Question:8 Solve the following system of inequalities graphically: x + y \leq 9, y > x, x\geq 0

    Answer:

    x + y \leq 9, y > x, x\geq 0

    Graphical representation of x+y=9,x=y and x=0 is given in graph below.

    For x + y \leq 9 ,

    The solution to this inequality is region below line (x+y=9) including points on this line because points on line also satisfy the inequality.

    For y > x ,

    The solution to this inequality represents half plane corresponding to the line (x=y) containing point (0,1) excluding points on this line because points on line does not satisfy the inequality.

    For x\geq 0 ,

    The solution to this inequality is region on right hand side of the line (x=0) 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:

    1635763196635

    Question:9 Solve the following system of inequalities graphically: 5x+4y\leq20, \ x\geq 1, \ y\geq 2

    Answer:

    5x+4y\leq20, \ x\geq 1, \ y\geq 2

    Graphical representation of \, ,5x+4y=20,\, \, \, x=1\, \, and \, \, y=2 is given in graph below.

    For 5x+4y\leq20, ,

    The solution to this inequality is region below the line (5x+4y=20) including points on this line because points on line also satisfy the inequality.

    For \ x\geq 1, ,

    The solution to this inequality is region right hand side of the line (x=1) including points on this line because points on line also satisfy the inequality.

    For \ y\geq 2,

    The solution to this inequality is region above the line (y=2) 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:

    1635763228699

    Question:10 Solve the following system of inequalities graphically: 3x + 4y \leq 60,\ x + 3y \leq 30, \ x \geq 0, \ y\geq 0

    Answer:

    3x + 4y \leq 60,\ x + 3y \leq 30, \ x \geq 0, \ y\geq 0

    Graphical representation of 3x+4y=60 \, \, ,x+3y=30\, \, \, ,x=0\, \, and\, \, y=0 is given in graph below.

    For 3x + 4y \leq 60 ,

    The solution to this inequality is region below the line (3x+4y=60) including points on this line because points on line also satisfy the inequality.

    For \ x + 3y \leq 30 ,

    The solution to this inequality is region below the line (x+3y=30) including points on this line because points on line also satisfy the inequality.

    For \ x \geq 0,

    The solution to this inequality is region right hand side of the line (x=0) including points on this line because points on line also satisfy the inequality.

    For \ y \geq 0,

    The solution to this inequality is region above the line (y=0) 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:

    1635763315671

    Question:11 Solve the following system of inequalities graphically: 2x +y \geq 4, \ x + y \leq 3, \ 2x - 3y \leq 6

    Answer:

    2x +y \geq 4, \ x + y \leq 3, \ 2x - 3y \leq 6

    Graphical representation of 2x+y=4 \, \, ,x+y=3 and 2x-3y=6 is given in graph below.

    For 2x +y \geq 4, ,

    The solution to this inequality is region above the line (2x+y=4) including points on this line because points on line also satisfy the inequality.

    For \ x + y \leq 3, ,

    The solution to this inequality is region below the line (x+y=3) including points on this line because points on line also satisfy the inequality.

    For \ 2x - 3y \leq 6,

    The solution to this inequality is region above the line (2x-3y= 6) 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:

    1635763400655

    Question:12 Solve the following system of inequalities graphically: x -2y \leq 3, 3x + 4y \geq 12, x \geq 0, y\geq 1

    Answer:

    x -2y \leq 3, 3x + 4y \geq 12, x \geq 0, y\geq 1

    Graphical representation of x-2y=3 \, \, ,3x+4y=12\, \, \, ,x=0\, \, and\, \, y=1 is given in graph below.

    For x -2y \leq 3 ,

    The solution to this inequality is region above the line (x-2y=3) including points on this line because points on line also satisfy the inequality.

    For 3x + 4y \geq 12 ,

    The solution to this inequality is region above the line (3x+4y=12) including points on this line because points on line also satisfy the inequality.

    For \ x \geq 0,

    The solution to this inequality is region right hand side of the line (x=0) including points on this line because points on line also satisfy the inequality.

    For \ y \geq 1,

    The solution to this inequality is region above the line (y=1) 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:

    1635763420814

    Question:13 Solve the following system of inequalities graphically: 4x + 3y \leq 60,\ y\geq 2x,\ x\geq 3,\ x,y\geq 0

    Answer:

    4x + 3y \leq 60,\ y\geq 2x,\ x\geq 3,\ x,y\geq 0

    Graphical representation of 4x+3y=60 \, \, ,y=2x\, \, \,,x=3\, \, ,x=0\, \, and\, \, y=0 is given in graph below.

    For 4x + 3y \leq 60,

    The solution to this inequality is region below the line (4x+3y=60) including points on this line because points on the line also satisfy the inequality.

    For y\geq 2x ,

    The solution to this inequality is region above the line (y=2x) including points on this line because points on the line also satisfy the inequality.

    For x\geq 3 ,

    The solution to this inequality is region right hand side of the line (x=3) including points on this line because points on the line also satisfy the inequality.

    For \ x \geq 0,

    The solution to this inequality is region right hand side of the line (x=0) including points on this line because points on the line also satisfy the inequality.

    For \ y \geq 0,

    The solution to this inequality is region above the line (y=0) 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:

    1635763491115

    Question:14 Solve the following system of inequality graphically: 3x + 2y \leq 150, \ x +4y \leq 80,\ x\leq 15 \ y\geq 0, \ x\geq 0

    Answer:

    3x + 2y \leq 150, \ x +4y \leq 80,\ x\leq 15 \ y\geq 0, \ x\geq 0

    Graphical representation of 3x+2y=150 \, \, ,x+4y=80\, \, \,,x=15\, \, ,x=0\, \, and\, \, y=0 is given in graph below.

    For 3x + 2y \leq 150,

    The solution to this inequality is region below the line (3x+2y=150) including points on this line because points on the line also satisfy the inequality.

    For x+4y\leq 80 ,

    The solution to this inequality is region below the line (x+4y=80) including points on this line because points on the line also satisfy the inequality.

    For x\leq 15 ,

    The solution to this inequality is region left hand side of the line (x=15) including points on this line because points on the line also satisfy the inequality.

    For \ x \geq 0,

    The solution to this inequality is region right hand side of the line (x=0) including points on this line because points on the line also satisfy the inequality.

    For \ y \geq 0,

    The solution to this inequality is region above the line (y=0) 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:

    1635763512601

    Question:15 Solve the following system of inequality graphically: x+2y \leq 10, \ x +y \geq 1, \ x-y\leq 0, x\geq 0, \ y\geq 0

    Answer:

    x+2y \leq 10, \ x +y \geq 1, \ x-y\leq 0, x\geq 0, \ y\geq 0

    Graphical representation of x+2y=10 \, \, ,x+y=1\, \, \,,x-y=0\, \, ,x=0\, \, and\, \, y=0 is given in graph below.

    For x+2y \leq 10,

    The solution to this inequality is region below the line (x+2y=10) including points on this line because points on line also satisfy the inequality.

    For \ x +y \geq 1, ,

    The solution to this inequality is region above the line (x+y=1) including points on this line because points on line also satisfy the inequality.

    For \ x-y\leq 0, ,

    The solution to this inequality is region above the line (x-y=0) including points on this line because points on line also satisfy the inequality.

    For \ x \geq 0,

    The solution to this inequality is region right hand side of the line (x=0) including points on this line because points on line also satisfy the inequality.

    For \ y \geq 0,

    The solution to this inequality is region above the line (y=0) 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:

    1635763543684


    Linear inequalities equations ncert solutions - Miscellaneous Exercise

    Question:1 Solve the inequality 2\leq 3x-4\leq5

    Answer:

    Given : 2\leq 3x-4\leq5


    2\leq 3x-4\leq5

    \Rightarrow\, \, 2+4\leq 3x\leq 5+4

    \Rightarrow\, \, 6\leq 3x\leq 9

    \Rightarrow\, \, \frac{6}{3}\leq x\leq \frac{9}{3}

    \Rightarrow\, \, 2\leq x\leq 3

    Thus, all the real numbers greater than equal to 2 and less than equal to 3 are solutions to this inequality.

    Solution set is \left \[2,3 \right \]

    Question:2 Solve the inequality 6 \leq -3(2x - 4) < 12

    Answer:

    Given 6 \leq -3(2x - 4) < 12

    6 \leq -3(2x - 4) < 12

    \Rightarrow\, \ \frac{6}{3}\leq -(2x-4)< \frac{12}{3}

    \Rightarrow\, \ -2\geq (2x-4)> -4

    \Rightarrow\, \ -2+4\geq 2x> -4+4

    \Rightarrow\, \ 2\geq 2x> 0

    \Rightarrow\, \ 1\geq x> 0

    Solution set is (01]

    Question:3 Solve the inequality -3 \leq 4 - \frac{7x}{2}\leq 18

    Answer:

    Given -3 \leq 4 - \frac{7x}{2}\leq 18


    \Rightarrow \, \, -3 \leq 4 - \frac{7x}{2}\leq 18

    \Rightarrow \, \, -3-4 \leq - \frac{7x}{2}\leq 18-4

    \Rightarrow \, \, -7 \leq - \frac{7x}{2}\leq 14

    \Rightarrow \, \, 7 \geq \frac{7x}{2} \geq -14

    \Rightarrow \, \, 7\times 2 \geq 7x\geq -14\times 2

    \Rightarrow \, \, 14 \geq 7x \geq -28

    \Rightarrow \, \, \frac{14}{7} \geq x \geq \frac{-28}{7}

    \Rightarrow \, \, 2 \geq x \geq -4

    Solution set is [-4,2]

    Question:4 Solve the inequality -15 < \frac{3(x-2)}{5} \leq 0

    Answer:

    Given The inequality

    -15 < \frac{3(x-2)}{5} \leq 0


    -15 < \frac{3(x-2)}{5} \leq 0

    \Rightarrow\, \ -15\times 5< 3(x-2)\leq 0\times 5

    \Rightarrow\, \ -75< 3(x-2)\leq 0

    \Rightarrow\, \ \frac{-75}{3}< (x-2)\leq \frac{0}{3}

    \Rightarrow\, \ -25< (x-2)\leq 0

    \Rightarrow\, \ -25+2< x\leq 0+2

    \Rightarrow\, \ -23< x\leq 2

    The solution set is (-23,2]

    Question:5 Solve the inequality -12<4-\frac{3x}{-5} \leq 2

    Answer:

    Given the inequality

    -12<4-\frac{3x}{-5} \leq 2


    -12<4-\frac{3x}{-5} \leq 2

    \Rightarrow\, \, -12-4< -\frac{3x}{-5}\leq 2-4

    \Rightarrow\, \, -16< -\frac{3x}{-5}\leq -2

    \Rightarrow\, \, -16< \frac{3x}{5}\leq -2

    \Rightarrow\, \, -16\times 5< 3x\leq -2\times 5

    \Rightarrow\, \, -80< 3x\leq -10

    \Rightarrow\, \, \frac{-80}{3}< 3x\leq \frac{-10}{3}

    Solution set is (\frac{-80}{3}, \frac{-10}{3}]

    Question:6 Solve the inequality 7 \leq \frac{(3x+ 11)}{2}\leq 11

    Answer:

    Given the linear inequality

    7 \leq \frac{(3x+ 11)}{2}\leq 11

    7 \leq \frac{(3x+ 11)}{2}\leq 11

    \Rightarrow \, \, 7\times 2 \leq (3x+ 11)\leq 11\times 2

    \Rightarrow \, \, 14 \leq (3x+ 11)\leq 22

    \Rightarrow \, \, 14-11 \leq (3x)\leq 22-11

    \Rightarrow \, \, 3 \leq 3x\leq 11

    \Rightarrow \, \, 1 \leq x\leq \frac{11}{3}

    The solution set of the given inequality is [1,\frac{11}{3}]

    Question:7 Solve the inequality and represent the solution graphically on number line. 5x + 1 > -24,\ 5x - 1 <24

    Answer:

    Given : 5x + 1 > -24,\ 5x - 1 <24


    5x + 1 > -24\, \, \, \, \, \, \, and\, \, \, \, \, \, \ 5x - 1 <24

    \Rightarrow 5x > -24-1\, \, \, \, \, \, \, and\, \, \, \, \, \, \ 5x <24+1

    \Rightarrow 5x > -25\, \, \, \, \, \, \, and\, \, \, \, \, \, \ 5x <25

    \Rightarrow x > \frac{-25}{5}\, \, \, \, \, \, \, and\, \, \, \, \, \, \ x <\frac{25}{5}

    \Rightarrow x > -5\, \, \, \, \, \, \, and\, \, \, \, \, \, \ x <5

    (-5,5)

    The solution graphically on the number line is as shown :

    1635763658335

    Question:8 Solve the inequality and represent the solution graphically on number line. 2(x-1)<x+5,\ 3(x+2)> 2 -x

    Answer:

    Given : 2(x-1)<x+5,\ 3(x+2)> 2 -x


    2(x-1)<x+5\, \, \, \, and\, \, \, \, \, \ 3(x+2)> 2 -x

    \Rightarrow \, \, 2x-2<x+5\, \, \, \, and\, \, \, \, \, \ 3x+6> 2 -x

    \Rightarrow \, \, 2x-x<2+5\, \, \, \, and\, \, \, \, \, \ 3x+x> 2 -6

    \Rightarrow \, \, x<7\, \, \, \, and\, \, \, \, \, \ 4x> -4

    \Rightarrow \, \, x<7\, \, \, \, and\, \, \, \, \, \ x> -1

    (-1,7)

    The solution graphically on the number line is as shown :

    1635763681053

    Question:9 Solve the inequality and represent the solution graphically on number line. 3x - 7 > 2(x-6),\ 6-x > 11 - 2x

    Answer:

    Given : 3x - 7 > 2(x-6),\ 6-x > 11 - 2x


    3x - 7 > 2(x-6)\, \, \, \, and\, \, \, \, \, \ 6-x > 11 - 2x

    \Rightarrow \, \, 3x - 7 > 2x-12\, \, \, \, and\, \, \, \, \, \ 6-x > 11 - 2x

    \Rightarrow \, \, 3x - 2x >7-12\, \, \, \, and\, \, \, \, \, \ 2x-x > 11 - 6

    \Rightarrow \, \, x >-5\, \, \, \, and\, \, \, \, \, \ x > 5

    x\in (5,\infty )

    The solution graphically on the number line is as shown :

    1635763701201

    Question:10 Solve the inequality and represent the solution graphically on number line.

    5(2x-7)-3(2x+3)\leq 0,\quad 2x + 19 \leq 6x +47

    Answer:

    Given : 5(2x-7)-3(2x+3)\leq 0,\quad 2x + 19 \leq 6x +47


    5(2x-7)-3(2x+3)\leq 0\, \, \, \, \, and\, \, \, \, \, \, \, \quad 2x + 19 \leq 6x +47

    \Rightarrow \, \, 10x-35-6x-9\leq 0\, \, \, \, \, and\, \, \, \, \, \, \, \quad 2x -6x\leq 47-19

    \Rightarrow \, \, 4x-44\leq 0\, \, \, \, \, and\, \, \, \, \, \, \, \quad -4x\leq 28

    \Rightarrow \, \, 4x\leq 44\, \, \, \, \, and\, \, \, \, \, \, \, \quad 4x\geq - 28

    \Rightarrow \, \, x\leq 11\, \, \, \, \, and\, \, \, \, \, \, \, \quad x\geq - 7

    x\in [-7,11]

    The solution graphically on the number line is as shown :

    1635763730673

    Question:11 A solution is to be kept between 68° F and 77° F. What is the range in temperature in degree Celsius (C) if the Celsius / Fahrenheit (F) conversion formula is given by F = \frac{9}{5}C + 32

    Answer:

    Since the solution is to be kept between 68° F and 77° F.

    68< F< 77

    Putting the value of F = \frac{9}{5}C + 32 , we have

    \Rightarrow \, \, \, 68< \frac{9}{5}C + 32< 77

    \Rightarrow \, \, \, 68-32< \frac{9}{5}C < 77-32

    \Rightarrow \, \, \, 36< \frac{9}{5}C < 45

    \Rightarrow \, \, \, 36\times 5< 9C < 45\times 5

    \Rightarrow \, \, \, 180< 9C < 225

    \Rightarrow \, \, \, \frac{180}{9}< C < \frac{225}{9}

    \Rightarrow \, \, \, 20< C < 25

    the range in temperature in degree Celsius (C) is between 20 to 25.

    Question:12 A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the 2% solution will have to be added?

    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.

    \therefore \, 2\%x+8\%\, of\, 640> 4\%\, of\, (640+x) and 2\%x+8\%\, of\, 640< 6\%\, of\, (x+640)

    \Rightarrow \, 2\%x+8\%\, of\, 640> 4\%\, of\, (640+x) and 2\%x+8\%\, of\, 640< 6\%\, of\, (x+640)

    \Rightarrow \, \frac{2}{100}x+(\frac{8}{100}) 640> \frac{4}{100} (640+x) \Rightarrow \, \frac{2}{100}x+(\frac{8}{100}) 640< \frac{6}{100} (640+x)

    \Rightarrow \, 2x+5120> 4x+2560 \Rightarrow \, 2x+5120< 6x+3840

    \Rightarrow \, 5120-2560> 4x-2x \Rightarrow \, 5120-3840< 6x-2x

    \Rightarrow \, 2560> 2x \Rightarrow \, 1280< 4x

    \Rightarrow \, 1280> x \Rightarrow \, 320< x

    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.

    Question:13 How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

    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.

    \therefore \, 30\%\, of\, (1125+x) > 45\%\, of\, (1125) and 25\%\, of\, (1125+x)< 45\%\, of\, 1125

    \Rightarrow \, 30\%\, of\, (1125+x) > 45\%\, of\, (1125) and 25\%\, of\, (1125+x)< 45\%\, of\, 1125

    \Rightarrow \, \frac{30}{100}(1125+x)> \frac{45}{100} (1125) \Rightarrow \, (\frac{25}{100}) (1125+x)< \frac{45}{100} (1125)

    \Rightarrow \, 30\times 1125+30x> 45\times (1125) \Rightarrow \, 25 (1125+x)< 45(1125)

    \Rightarrow \, 30x> (45-30)\times (1125) \Rightarrow \, 25 x< (45-25)1125

    \Rightarrow \, 30x> (15)\times (1125) \Rightarrow \, 25 x< (20)1125

    \Rightarrow \, x> \frac{15\times 1125}{30} \Rightarrow \, x< \frac{20\times 1125}{25}

    \Rightarrow \, x> 562.5 \Rightarrow \, x< 900

    Thus, the number of litres water that is to be added will have to be more than 562.5 and less than 900 litres.

    Question:14 IQ of a person is given by the formula IQ= \frac{MA}{CA}\times 100 where MA is mental age and CA is chronological age. If 80\leq IQ\leq140 for a group of 12 years old children, find the range of their mental age.

    Answer:

    Given that group of 12 years old children.

    80\leq IQ\leq140

    For a group of 12 years old children, CA =12 years

    IQ= \frac{MA}{CA}\times 100

    Putting the value of IQ, we obtain

    80\leq IQ\leq140

    \Rightarrow \, \, 80\leq \frac{MA}{CA}\times 100\leq140

    \Rightarrow \, \, 80\leq \frac{MA}{12}\times 100\leq140

    \Rightarrow \, \, 80\times 12\leq MA\times 100\leq140\times 12

    \Rightarrow \, \, \frac{80\times 12}{100}\leq MA\leq \frac{140\times 12}{100}

    \Rightarrow \, \, 9.6\leq MA\leq 16.8

    Thus, the range of mental age of the group of 12 years old children is \, \, 9.6\leq MA\leq 16.8

    Class 11 maths chapter 6 NCERT solutions - Symmary

    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.

    Linear Inequality 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

    \\9x+12y\leq 180 \(fabricating \ constraint)\\x+3y\leq 30\ (finishing\ constraint)

    Linear Inequalities Exercise Wise Solutions

    NCERT Solutions For Class 11 Mathematics - Chapter Wise

    Key Features Of Linear Inequalities Class 11 Maths NCERT Chapter

    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".

    NCERT Solutions For Class 11- Subject Wise

    Benefits of NCERT Solutions

    • Linear inequalities equations ncert solutions 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 class 11 maths ch 6 question answer 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.
    • NCERT solutions for maths chapter 6 class 11 will some innovative ways of solving the problems which become very important to solve some specific problems in an easy way.
    • This ch 6 maths class 11 also useful in the prediction of future events based on the past data which is the fundamentals of machine learning

    NCERT Books and NCERT Syllabus

    Happy Reading !!!

    Frequently Asked Question (FAQs)

    1. What are important topics of the chapter Linear Inequalities ?

    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.

    2. Explain the steps to plot a graph of linear inequality covered in NCERT Solutions for Class 11 Maths 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

    • Label the graph

    Students can find NCERT solutions for class 11 maths  by clicking on the link.

    3. List out the number of exercises present in NCERT solutions for class 11 maths chapter 6.

    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

    4. Which is the official website of NCERT ?

    NCERT official is the official website of the NCERT where you can get NCERT textbooks and syllabus from class 1 to 12.

    Articles

    Get answers from students and experts

    A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

    Option 1)

    0.34\; J

    Option 2)

    0.16\; J

    Option 3)

    1.00\; J

    Option 4)

    0.67\; J

    A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

    Option 1)

    2.45×10−3 kg

    Option 2)

     6.45×10−3 kg

    Option 3)

     9.89×10−3 kg

    Option 4)

    12.89×10−3 kg

     

    An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

    Option 1)

    2,000 \; J - 5,000\; J

    Option 2)

    200 \, \, J - 500 \, \, J

    Option 3)

    2\times 10^{5}J-3\times 10^{5}J

    Option 4)

    20,000 \, \, J - 50,000 \, \, J

    A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

    Option 1)

    K/2\,

    Option 2)

    \; K\;

    Option 3)

    zero\;

    Option 4)

    K/4

    In the reaction,

    2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

    Option 1)

    11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

    Option 2)

    6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

    Option 3)

    33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

    Option 4)

    67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

    How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

    Option 1)

    0.02

    Option 2)

    3.125 × 10-2

    Option 3)

    1.25 × 10-2

    Option 4)

    2.5 × 10-2

    If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

    Option 1)

    decrease twice

    Option 2)

    increase two fold

    Option 3)

    remain unchanged

    Option 4)

    be a function of the molecular mass of the substance.

    With increase of temperature, which of these changes?

    Option 1)

    Molality

    Option 2)

    Weight fraction of solute

    Option 3)

    Fraction of solute present in water

    Option 4)

    Mole fraction.

    Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

    Option 1)

    twice that in 60 g carbon

    Option 2)

    6.023 × 1022

    Option 3)

    half that in 8 g He

    Option 4)

    558.5 × 6.023 × 1023

    A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

    Option 1)

    less than 3

    Option 2)

    more than 3 but less than 6

    Option 3)

    more than 6 but less than 9

    Option 4)

    more than 9

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    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.

    3 Jobs Available
    Vlogger

    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 cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

    3 Jobs Available
    Travel Journalist

    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.

    2 Jobs Available
    Videographer

    Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production. 

    2 Jobs Available
    SEO Analyst

    An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns. 

    2 Jobs Available
    Product Manager

    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.  

    3 Jobs Available
    Quality Controller

    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.

    3 Jobs Available
    Production Manager

    Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers. 

    Resource Links for Online MBA 

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    QA Manager

    Quality Assurance Manager Job Description: A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes. 

    2 Jobs Available
    QA Lead

    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. 

    2 Jobs Available
    Reliability Engineer

    Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment. 

    2 Jobs Available
    Safety Manager

    A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

    2 Jobs Available
    Corporate Executive

    Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

    2 Jobs Available
    Information Security Manager

    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 

    3 Jobs Available
    Computer Programmer

    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.

    3 Jobs Available
    Product Manager

    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.  

    3 Jobs Available
    ITSM Manager

    ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks. 

    3 Jobs Available
    .NET Developer

    .NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of  creating, designing and developing applications using .NET languages such as VB and C#. 

    2 Jobs Available
    Corporate Executive

    Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

    2 Jobs Available
    DevOps Architect

    A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties. 

    2 Jobs Available
    Cloud Solution Architect

    Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.

    2 Jobs Available
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