Linear Inequalities (Ex – 6.2)
Solve the following inequalities graphically in two-dimensional plane
Question 1.
x + y < 5
Solution.
Consider the equation x + y = 5. It passes through the points (0, 5) and (5, 0). The line x + y = 5 is represented by AB. Consider the inequality x + y < 5
Put x = 0, y = 0
0 + 0 = 0 < 5, which is true. So, the origin O lies in the plane x + y < 5
∴ Shaded region represents the inequality x + y < 5
Question 2.
2x + y ≥ 6
Solution.
Consider the equation 2x + y = 6
The line passes through (0, 6), (3, 0).
The line 2x + y = 6 is represented by AB.
Now, consider 2x + y ≥ 6
Put x = 0, y = 0
0 + 0 ≥ 6, which does not satisfy this inequality.
∴ Origin does not lie in the region of 2x + y ≥ 6.
The shaded region represents the inequality 2x + y ≥ 6
Question 3.
3x + 4y ≤ 12
Solution.
We draw the graph of the equation 3x + 4y = 12. The line passes through the points (4, 0), (0, 3). This line is represented by AB. Now consider the inequality 3x + 4y ≤ 12
Putting x = 0, y = 0 0 + 0 = 0 ≤ 12, which is true
∴ Origin lies in the region of 3x + 4y ≤ 12 The shaded region represents the inequality 3x + 4y ≤ 12
Question 4.
y + 8 ≥ 2x
Solution.
Given inequality is y + 8 ≥ 2x
Let us draw the graph of the line, y+ 8 = 2x
The line passes through the points (4, 0), (0, -8).
This line is represented by AB.
Now, consider the inequality y + 8 ≥ 2x.
Putting x = 0, y = 0
0 + 8 ≥ 0, which is true
∴ Origin lies in the region of y + 8 ≥ 2x
The shaded region represents the inequality y + 8 ≥ 2x.
Question 5.
x – y ≤ 2
Solution.
Given inequality is x – y ≤ 2
Let us draw the graph of the line x – y = 2
The line passes through the points (2, 0), (0, -2)
This line is represented by AB.
∴ Origin lies in the region of x – y ≤ 2
The shaded region represents the inequality x – y ≤ 2.
Question 6.
2x – 3y > 6
Solution.
We draw the graph of line 2x – 3y = 6.
The line passes through (3, 0), (0, -2)
AB represents the equation 2x – 3y = 6
Now consider the inequality 2x – 3y > 6
Putting x = 0, y = 0
0 – 0 > 6, which is not true
∴ Origin does not lie in the region of 2x – 3y > 6.
The shaded region represents the inequality 2x – 3y > 6
Question 7.
-3x + 2y ≥ -6.
Solution.
Let us draw the line -3x + 2y = -6
The line passes through (2, 0), (0, -3)
The line AB represents the equation -3x + 2y = -6
Now consider the inequality -3x+ 2y ≥ -6
Putting x = 0, y = 0
0 + 0 ≥ -6, which is true.
∴ Origin lies in the region of -3x + 2y ≥ -6
The shaded region represents the inequality -3x + 2y ≥ – 6
Question 8.
3y- 5x < 30
Solution.
Given inequality is 3y – 5x < 30
Let us draw the graph of the line 3y – 5x = 30
The line passes through (-6, 0), (0, 10)
The line AB represents the equation 3y – 5x = 30
Now, consider the inequality 3y – 5x < 30
Putting x = 0, y = 0
0 – 0 < 30, which is true.
∴ Origin lies in the region of 3y – 5x < 30
The shaded region represents the inequality 3y – 5x < 30
Question 9.
y<- 2
Solution.
Given inequality is y < -2 ………(1)
Let us draw the graph of the line y = -2
AB is the required line.
Putting y = 0 in (1), we have
0 < -2, which is not true.
The solution region is the shaded region below the line.
Hence, every point below the line (excluding the line) is the solution area.
Question 10.
x > -3
Solution.
Let us draw the graph of x = -3
∴ AB represents the line x = -3
By putting x = 0 in the inequality x > -3
We get, 0 > -3, which is true.
∴ Origin lies in the region of x > -3.
Graph of the inequality x > -3 is shown in the figure by the shaded area
