Chapter 6 Linear Inequalities Ex 6.3 Solutions
Question - 11 : - Solve the following system of inequalities graphically: 2x + y≥ 4, x + y ≤ 3, 2x – 3y ≤ 6
Answer - 11 : -
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).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question - 12 : - Solve the following system of inequalities graphically:x – 2y ≤3, 3x + 4y ≥ 12, x ≥ 0, y ≥1
Answer - 12 : -
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).
Hence, 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 follows.
Question - 13 : - Solve the following system of inequalities graphically:
4x + 3y ≤ 60, y ≥ 2x, x ≥ 3, x, y ≥ 0
Answer - 13 : -
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).
Hence, the solution of the given system of linear inequalities is represented by the common shaded region including the points on the respective lines as follows.
Question - 14 : - Solve the following system of inequalities graphically: 3x + 2y ≤ 150, x + 4y ≤ 80, x ≤ 15, y ≥ 0, x ≥ 0
Answer - 14 : -
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.
Question - 15 : - Solve the following system of inequalities graphically: x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0
Answer - 15 : -
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.