Graph the solution set to the inequality. 2x+3y>12
graph the line 2x+3y=12
intercepts at (6,0)and (4,0)
to check which region to shade, subs in a point i.e. (0,0) 0 is not greater than 12
shade the area on the opp side of the (0,0) which is separated by the line
hahatdog
To graph the solution set of the inequality 2x + 3y > 12, we first need to graph the boundary line of the inequality, which is 2x + 3y = 12.
To graph this line, we can rewrite it in slope-intercept form by solving for y:
3y = -2x + 12
y = (-2/3)x + 4
Using this equation, we can plot two points on the line:
Let x = 0:
y = (-2/3)(0) + 4
y = 4
So one point on the line is (0, 4).
Let x = 3:
y = (-2/3)(3) + 4
y = 2
So another point on the line is (3, 2).
Plotting these two points and drawing a straight line passing through them, we get the boundary line:
|
4 | *
|
2 | *
|
|__________________________
0 2 4 6 8 10 12 14 16 ...
Since the inequality is greater than, we need to determine which side of the line represents the solution set. To do this, we can pick a point outside of the line and substitute it into the inequality. For example, let's use the point (0, 0):
2(0) + 3(0) > 12
0 + 0 > 12
0 > 12
Since the inequality is not true, we know that (0, 0) is not part of the solution set. Therefore, we shade the side of the line that does not include the origin:
|
4 | *
| * |
2 | * |
| \ |
|__________________________
0 2 4 6 8 10 12 14 16 ...
This shaded region represents the solution set to the inequality 2x + 3y > 12.
To graph the solution set to the inequality 2x + 3y > 12, we need to first graph the equation 2x + 3y = 12, which is the boundary line of the inequality.
To graph the line 2x + 3y = 12, we can start by finding two points that lie on the line. To do this, we can assign values to x and solve for y, and vice versa. Let's choose x = 0:
When x = 0, the equation becomes 2(0) + 3y = 12, which simplifies to 3y = 12. Dividing both sides by 3, we get y = 4. So, one point on the line is (0, 4).
Now, let's choose y = 0:
When y = 0, the equation becomes 2x + 3(0) = 12, which simplifies to 2x = 12. Dividing both sides by 2, we get x = 6. So, another point on the line is (6, 0).
Now, plot these two points on a coordinate plane and draw a straight line passing through them.
Once we have the line 2x + 3y = 12 graphed, we need to determine which side of the line satisfies the inequality 2x + 3y > 12. To do this, we can choose a test point not on the line and substitute its coordinates into the inequality. For simplicity, let's choose (0, 0) as the test point.
Substituting x = 0 and y = 0 into the inequality, we get 2(0) + 3(0) > 12, which simplifies to 0 > 12. Since this is false, the side of the line NOT containing the test point (0, 0) satisfies the inequality.
Therefore, the solution set to the inequality 2x + 3y > 12 is the entire region above the line 2x + 3y = 12 on the graph.