To obtain the common region for both given linear inequalities, we have to follow the steps given below.
Step 1 :
First, we have to assume the given inequalities as equations.
Then, we have to draw the graph of the line using x and y-intercepts.
Step 2 :
The region which satisfies the inequality can be shaded.
To check if the inequality is being satisfied by applying any point from the region, the point which satisfies the inequality is known as the solution region and that can be shaded.
Step 3 :
Do the process explained in step 2 for the second inequality.
Step 4 :
The overlapping region is the solution region for the system of inequalities given.
Sketch the graph of the system of linear inequalities.
Example 1 :
2x + 1 > 0
3y - 6 > 0
Solution :
Given,
2x + 1 > 0 and 3y - 6 > 0
Converting the inequalities into equations.
2x + 1 = 0 and 3y - 6 = 0
2x + 1 = 0 2x = -1 x = -1/2 |
3y - 6 = 0 3y = 6 y = 2 |
By plotting the x and y values on the graph, we get
If x = 0, then 2(0) + 1 > 0 (True)
So, we can shade the region which is right to 2x + 1 > 0
If y = 3, then 3(3) - 6 > 0 (True)
So, we can shade the region which is above to 3y - 6 > 0
Example 2 :
2x + 3y ≥ 6
4x + 6y ≤ 24
Solution :
Given,
2x + 3y ≥ 6 and 4x + 6y ≤ 24
Converting the inequalities into equations.
2x + 3y = 6 and 4x + 6y = 24
To graph the line 2x + 3y = 6, we find x and y intercepts.
x-intercept : Put y = 0 2x + 3(0) = 6 2x = 6 x = 3 |
y-intercept : Put x = 0 2(0) + 3y = 6 3y = 6 y = 2 |
To graph the line 4x + 6y = 24, we find x and y intercepts.
x-intercept : Put y = 0 4x + 6(0) = 24 4x = 24 x = 6 |
y-intercept : Put x = 0 4(0) + 6y = 24 6y = 24 y = 4 |
The point on the line 2x + 3y = 6 are (3, 0) and (0, 2)
The point on the line 4x + 6y = 24 are (6, 0) and (0, 4)
By plotting the points on the graph, we get
If x = 4 and y = 3, then 2(4) + 3(3) ≥ 6 (True)
So, we can shade the region which is above to 2x + 3y ≥ 6.
If x = 2 and y = 2, then 4(2) + 6(2) ≤ 24 (True)
So, we can shade the region which is below to 4x + 6y ≤ 24.
Example 3 :
2y - x ≥ -6
2y - 3x < -6
Solution :
Given,
2y - x ≥ -6 and 2y - 3x < -6
Converting the inequalities into equations.
2y - x = -6 and 2y - 3x = -6
To graph the line 2y - x = -6, we find x and y intercepts.
x-intercept : Put y = 0 2y - x = -6 2(0) - x = -6 - x = -6 x = 6 |
y-intercept : Put x = 0 2y - x = -6 2y - 0 = -6 2y = -6 y = -3 |
To graph the line 2y - 3x = -6, we find x and y intercepts.
x-intercept : Put y = 0 2y - 3x = -6 2(0) - 3x = -6 - 3x = -6 x = 2 |
y-intercept : Put x = 0 2y - 3x = -6 2y - 3(0) = -6 2y = -6 y = -3 |
The point on the line 2y - x = -6 are (6, 0) and (0, -3)
The point on the line 2y - 3x = -6 are (2, 0) and (0, -3)
By plotting the points on the graph, we get
If x = 3 and y = 0, then 2(0) - 3 ≥ -6 (True)
So, we can shade the region which is above to 2y - x ≥ -6.
If x = 3 and y = -5, then 2(-5) - 3(3) < -6 (True)
So, we can shade the region which is below to 2y - 3x < -6
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