Referring to the AP Inter 1st Year Maths Study Material Chapter 5 Linear Inequalities Exercise 5c Solutions makes it easier to understand complex problems.
AP Inter 1st Year Maths Linear Inequalities Solutions Exercise 5c
Solve the following inequalities graphically in two-dimensional plane,
Question 1.
x + y < 5.
Solution:
The graphical representation of x + y = 5 is given as dotted line in the figure. This line divides the xy-plane in two planes I and II.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that,
0 + 0 < 5 or 0 < 5, which is true.
Therefore, half-plane II is not the solution region of the given inequality. Also, it is evident that any point on the line does not satisfy the given strict inequality.
Thus, the solution region of the given inequality is the shaded half plane I excluding the points on the line.
This can be represented as shown in graph.
Question 2.
2x + y ≥ 6.
Solution:
The graphical representation of 2x + y = 6 is given in the figure. This line divides the xy-plane in two half planes, I and II.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 2(0) + 0 ≥ 6 or 0 ≥ 6, which is false.
Therefore, half plane I is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the shaded half plane II including the points on the line.
This can be represented as shown in graph.
Question 3.
3x + 4y ≤ 12.
Solution:
The graphical representation of 3x + 4y = 12 is given in the figure. This line divides the xy-plane in two half planes, I and II.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 3(0) + 4(0) < 12 or 0 < 12, which is true.
Thus, the solution region of the given in equality is the shaded half plane I, including the points on the line.
This can be represented as shown in graph.
Question 4.
y + 8 ≥ 2x.
Solution:
The graphical representation of y + 8 = 2x is given in the figure. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 0 + 8 ≥ 2(0) or 8 ≥ 0, which is true.
Therefore, lower half plane is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the upper half plane containing the points including the line.
The solution region is represented by the shaded region as shown in graph.
Question 5.
x – y ≤ 2.
Solution:
The graphical representation of x – y = 2 is given in the figure. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 0 – 0 ≤ 2 or 0 ≤ 2, which is true.
Therefore, the lower half plane is not the solution region of the given inequality. Also, any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the upper half plane containing the points including the line.
The solution region is represented by the shaded region as shown in graph.
Question 6.
2x – 3y > 6.
Solution:
The graphical representation of 2x – 3y = 6 is given as dotted line in the figure. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 2(0) – 3(0) ≥ 6 or 0 ≥ 6, which is false.
Therefore, the upper half plane is not the solution region of the given inequality. Also, any point on the line does not satisfy the given inequality.
Thus, the solution region of the given inequality is the lower half plane that does not contain the points including the line.
The solution region is represented by the shaded region as shown in graph.
Question 7.
– 3x + 2y ≥ – 6.
Solution:
The graphical representation of -3x + 2y = -6 is given in the figure. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that -3(0) + 2(0) ≥ -6 or 0 ≥ – 6, which is true. .
Therefore, the lower half plane is not the solution region of the given inequality. Also, it is evident that any point on the line satisfies the given inequality.
Thus, the solution region of the given inequality is the upper half plane containing the points including the line.
The solution region is represented by the shaded region as shown in graph.
Question 8.
3y – 5x < 30.
Solution:
The graphical representation of 3y – 5x = 30 is given as dotted line in the figure. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 3(0) – 5(0) < 30 or 0 < 30, which is true.
Therefore, the upper half plane is not the solution region of the given inequality. Also, it is evident that any point on the line does not satisfy the given inequality.
Thus, the solution region of the given inequality is the lower half plane containing the points excluding the line.
The solution region is represented by the shaded region as shown in graph.
Question 9.
y < -2.
Solution:
The graphical representation of y = – 2 is given as dotted line in the figure. This line divides the xiy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 0 < -2, which is false.
Also, it is evident that any point on the line does not satisfy the given inequality.
Hence, every point below the line, y = – 2 (excluding all the points on the line), determine the solution of the given inequality.
The solution region is represented by the shaded region as shown in graph,
Question 10.
x > – 3.
Solution:
The graphical representation of x = – 3 is given as dotted line in the figure. This line divides the xy-plane in two half planes.
Select a point (not on the line), which lies in one of the half planes, to determine whether the point satisfies the given inequality or not.
We select the point as (0, 0).
It is observed that, 0 > -3, which is true.
Also, it is evident that any point on the line does not satisfy the given inequality.
Hence, every point on the right side of the line, x= -3 (excluding all the points on the line), determines the solution of the given inequality.