**NCERT Solutions for Class 11 Maths Chapter 6 Linear Inequalities Ex 6.3**

NCERT Solutions for Class
11 Maths Chapter 6 Linear
Inequalities Ex 6.3 are the part of NCERT Solutions for Class 11 Maths. Here
you can find the NCERT Solutions for Class 11 Maths chapter 6 Linear
Inequalities Ex 6.3.

**Solve the following system of inequalities
graphically:
**

**Ex 6.3 Class 11 Maths Question 1.**

x ≥ 3, y ≥ 2

**Solution:**

x ≥ 3 … (1)

y
≥ 2 … (2)

The
graph of the lines, x = 3 and y = 2, are drawn in the figure below.

Inequality
(1) represents the region on the right hand side of the line, x = 3 (including
the line x = 3), and inequality (2) represents the region above the line, y = 2
(including the line y = 2).

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region including the points on the respective lines as shown
below.

**Ex 6.3 Class 11 Maths Question
2.**

3x + 2y ≤ 12, x ≥ 1, y ≥ 2

**Solution:**

3x + 2y ≤ 12 … (1)

x
≥ 1 … (2)

y
≥ 2 … (3)

The
graphs of the lines, 3x + 2y = 12, x = 1 and y = 2, are drawn in the figure
below.

Inequality
(1) represents the region below the line, 3x + 2y = 12 (including the line 3x +
2y = 12). Inequality (2) represents the region on the right side of the line, x
= 1 (including the line x = 1). Inequality (3) represents the region above the
line, y = 2 (including the line y = 2).

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region including the points on the respective lines as shown
below.

**Ex 6.3 Class 11 Maths Question 3.**

2x + y ≥ 6, 3x + 4y ≤ 12

**Solution:**

2x + y ≥ 6 … (1)

3x
+ 4y ≤ 12 … (2)

The
graph of the lines, 2x + y = 6 and 3x + 4y = 12, are drawn in the figure below.

Inequality
(1) represents the region above the line, 2x + y= 6 (including the line 2x + y=
6), and inequality (2) represents the region below the line, 3x + 4y =12
(including the line 3x + 4y =12).

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region including the points on the respective lines as shown
below.

**Ex 6.3 Class 11 Maths Question 4.**

x + y ≥ 4, 2x – y > 0

**Solution:**

x + y ≥ 4 … (1)

2x
– y > 0 … (2)

The
graph of the lines, x + y = 4 and 2x – y = 0, are drawn in the figure below.

Inequality
(1) represents the region above the line, x + y = 4 (including the line x + y =
4).

It
is observed that (1, 0) satisfies the inequality, 2x – y > 0. [2(1) – 0 = 2
> 0]

Therefore,
inequality (2) represents the half plane corresponding to the line, 2x – y = 0,
containing the point (1, 0) [excluding the line 2x – y > 0].

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region including the points on line x + y = 4 and excluding the
points on line 2x – y = 0 as shown below.

**Ex 6.3 Class 11 Maths Question 5.**

2x – y > 1, x – 2y < -1

**Solution:**

2x – y > 1 … (1)

x
– 2y < –1 … (2)

The
graph of the lines, 2x – y = 1 and x – 2y = –1, are drawn in the figure below.

Inequality
(1) represents the region below the line, 2x – y = 1 (excluding the line 2x – y
= 1), and inequality (2) represents the region above the line, x – 2y = –1
(excluding the line x – 2y = –1).

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region excluding the points on the respective lines as shown
below.

**Ex 6.3 Class 11 Maths Question 6.**

x + y ≤ 6, x + y ≥ 4**Solution:**

x + y ≤ 6 … (1)

x
+ y ≥ 4 … (2)

The
graph of the lines, x + y = 6 and x + y = 4, are drawn in the figure below.

Inequality
(1) represents the region below the line, x + y = 6 (including the line x + y =
6), and inequality (2) represents the region above the line, x + y = 4
(including the line x + y = 4).

**Ex 6.3 Class 11 Maths Question 7.**

2x + y ≥ 8, x + 2y ≥ 10

**Solution:**

2x + y = 8 … (1)

x
+ 2y = 10 … (2)

The
graph of the lines, 2x + y = 8 and x + 2y = 10, are drawn in the figure below.

Inequality
(1) represents the region above the line, 2x + y = 8, and inequality (2)
represents the region above the line, x + 2y = 10.

**Ex 6.3 Class 11 Maths Question 8.**

x + y ≤ 9, y > x, x ≥ 0

**Solution:**

x + y ≤ 9 ... (1)

y
> x
... (2)

x
≥ 0 ...
(3)

The
graph of the lines, x + y = 9 and y = x, are drawn in the figure below.

Inequality
(1) represents the region below the line, x + y = 9 (including the line x + y =
9).

It
is observed that (0, 1) satisfies the inequality, y > x. [1 > 0]

Therefore,
inequality (2) represents the half plane corresponding to the line, y = x,
containing the point (0, 1) [excluding the line y = x].

Inequality
(3) represents the region on the right hand side of the line, x = 0 or y-axis
(including y-axis).

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region including the points on the lines, x + y = 9 and x = 0,
and excluding the points on line y = x as shown below.

**Ex 6.3 Class 11 Maths Question 9.**

5x + 4y ≤ 20, x ≥ 1, y ≥ 2

**Solution:**

5x + 4y ≤ 20 … (1)

x
≥ 1 … (2)

y
≥ 2 … (3)

The
graph of the lines, 5x + 4y = 20, x = 1 and y = 2, are drawn in the figure
below.

Inequality
(1) represents the region below the line, 5x + 4y = 20 (including the line 5x +
4y = 20). Inequality (2) represents the region on the right hand side of the
line, x = 1 (including the line x = 1). Inequality (3) represents the region
above the line, y = 2 (including the line y = 2).

**Ex 6.3 Class 11 Maths Question 10.**

3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0

**Solution:**

3x + 4y ≤ 60 … (1)

x
+ 3y ≤ 30 … (2)

The
graph of the lines, 3x + 4y = 60 and x + 3y = 30, are drawn in the figure
below.

Inequality
(1) represents the region below the line, 3x + 4y = 60 (including the line 3x +
4y = 60), and inequality (2) represents the region below the line, x + 3y = 30
(including the line x + 3y = 30).

Since
x ≥ 0 and y ≥ 0, every point in the common shaded region in the first quadrant
including the points on the respective line and the axes represents the
solution of the given system of linear inequalities.

**Ex 6.3 Class 11 Maths Question 11.**

2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6

**Solution:**

2x + y ≥ 4 … (1)

x
+ y ≤ 3 … (2)

2x
– 3y ≤ 6 … (3)

The
graph of the lines, 2x + y = 4, x + y = 3, and 2x – 3y = 6, are drawn in the
figure below.

Inequality
(1) represents the region above the line, 2x + y = 4 (including the line 2x + y
= 4). Inequality (2) represents the region below the line, x + y = 3 (including
the line x + y = 3). Inequality (3) represents the region above the line, 2x –
3y = 6 (including the line 2x – 3y = 6).

**Ex 6.3 Class 11 Maths Question 12.**

x – 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1

**Solution:**

x – 2y ≤ 3 … (1)

3x
+ 4y ≥ 12 … (2)

y
≥ 1 … (3)

The
graph of the lines, x – 2y = 3, 3x + 4y = 12 and y = 1, are drawn in the figure
below.

Inequality
(1) represents the region above the line, x – 2y = 3 (including the line x – 2y
= 3). Inequality (2) represents the region above the line, 3x + 4y = 12
(including the line 3x + 4y = 12). Inequality (3) represents the region above
the line, y = 1 (including the line y = 1).

The
inequality, x ≥ 0, represents the region on the right hand side of y-axis
(including y-axis).

Therefore,
the solution of the given system of linear inequalities is represented by the
common shaded region including the points on the respective lines and y- axis
as shown below.

**Ex 6.3 Class 11 Maths Question 13.**

4x + 3y ≤ 60, y ≥ 2x, x ≥ 3, x, y ≥ 0

**Solution:**

4x + 3y ≤ 60 … (1)

y
≥ 2x … (2)

x
≥ 3 … (3)

The
graph of the lines, 4x + 3y = 60, y = 2x and x = 3, are drawn in the figure
below.

Inequality
(1) represents the region below the line, 4x + 3y = 60 (including the line 4x +
3y = 60). Inequality (2) represents the region above the line, y = 2x
(including the line y = 2x). Inequality (3) represents the region on the right
hand side of the line, x = 3 (including the line x = 3).

**Ex 6.3 Class 11 Maths Question 14.**

3x + 2y ≤ 150, x + 4y ≤ 80, x ≤ 15, y ≥ 0, x ≥ 0

**Solution:**

3x + 2y ≤ 150 … (1)

x
+ 4y ≤ 80 … (2)

x
≤ 15 … (3)

The
graph of the lines, 3x + 2y = 150, x + 4y = 80 and x = 15, are drawn in the
figure below.

Inequality
(1) represents the region below the line, 3x + 2y = 150 (including the line 3x
+ 2y = 150). Inequality (2) represents the region below the line, x + 4y = 80
(including the line x + 4y = 80). Inequality (3) represents the region on the
left hand side of the line, x = 15 (including the line x = 15).

Since
x ≥ 0 and y ≥ 0, every point in the common shaded region in the first quadrant
including the points on the respective lines and the axes represents the
solution of the given system of linear inequalities.

**Ex 6.3 Class 11 Maths Question 15.**

x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0

**Solution:**

x + 2y ≤ 10 … (1)

x
+ y ≥ 1 … (2)

x
– y ≤ 0 … (3)

The
graph of the lines, x + 2y = 10, x + y = 1 and x – y = 0, are drawn in the
figure below.

Inequality
(1) represents the region below the line, x + 2y = 10 (including the line x +
2y = 10). Inequality (2) represents the region above the line, x + y = 1
(including the line x + y = 1). Inequality (3) represents the region above the
line, x – y = 0 (including the line x – y = 0).

Since
x ≥ 0 and y ≥ 0, every point in the common shaded region in the first quadrant
including the points on the respective lines and the axes represents the
solution of the given system of linear inequalities.

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