Select all of the ordered pairs that are solutions to the following system of inequalities.
2x+2y≥7
y<3x–5
To find the solutions to the system of inequalities, we need to find all the ordered pairs (x, y) that satisfy both inequalities simultaneously.
Let's start by finding the solutions to each inequality separately:
1. 2x + 2y ≥ 7
To find the solutions to this inequality, we can rewrite it as:
y ≥ (7 - 2x) / 2
Let's plot the solutions to this inequality on a graph:
y = (7 - 2x) / 2
This is equivalent to:
y = 3.5 - x
Now let's plot this line on the graph:
y = 3.5 - x
Next, we have to determine which side of the line represents the solutions to the inequality. We know that any point above the line is a solution, because it corresponds to y values larger than the y values on the line. So, we shade the region above the line.
2. y < 3x - 5
To find the solutions to this inequality, we can rewrite it as:
y < 3x - 5
Let's plot the solutions to this inequality on a graph:
y = 3x - 5
Now let's plot this line on the graph:
y = 3x - 5
We shade the region below the line, because any point below the line has y values lower than the values on the line.
Now, by looking at the overlapping shaded areas of the two inequalities, we can identify the solutions to the system of inequalities:
The ordered pairs that are solutions to the system of inequalities are the points that lie in the region where both shaded areas overlap.