NCERT Solutions for Exercise 6.2 Class 11 Maths Chapter 6 - Linear Inequalities

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:

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

Answer:

$2x+y\ge 6$

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

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

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

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:

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

Answer:

$3x+4y\le 12$

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

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

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

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:

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

Answer:

$y+8\ge 2x$

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

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

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

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:

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

Answer:

$x-y\le 2$

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

$1-2\le 2$ i.e. $-1\le 2$ , which is true.

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

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:

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:

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

Answer:

$-3x+2y\ge -6$

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

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

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

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:

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:

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:

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: