Which of the following points lie in the solution set to the following system of inequalities?
y < −3x + 3
y < x + 2
To find the solution set to this system of inequalities, we need to find all the points that satisfy both inequalities simultaneously.
One way to do this is to graph the two inequalities on the same coordinate plane and shade the region where the solutions overlap.
To graph the first inequality, y < −3x + 3, we can start by graphing the line y = −3x + 3. This line has a y-intercept of 3 and a slope of −3, so we can plot the point (0, 3) and then use the slope to find another point, say (1, 0), and draw the line connecting them.
Next, we need to determine which side of the line represents the solutions to the inequality. Since y is less than −3x + 3, we want the region below the line (excluding the line itself). We can shade this region to indicate the solutions:
graph{y < -3x + 3 [-10, 10, -5, 5]}
To graph the second inequality, y < x + 2, we can start by graphing the line y = x + 2. This line has a y-intercept of 2 and a slope of 1, so we can plot the point (0, 2) and then use the slope to find another point, say (1, 3), and draw the line connecting them.
Again, we need to determine which side of the line represents the solutions to the inequality. Since y is less than x + 2, we want the region below the line (excluding the line itself). We can shade this region to indicate the solutions:
graph{y < x + 2 [-10, 10, -5, 5]}
Now we can see that the solution set to the system of inequalities is the region where the shaded regions intersect. This region includes all the points that satisfy both inequalities.
To check which of the following points lie in the solution set, we can simply plot them on the same coordinate plane and see if they fall within the shaded region:
- (0, 0): This point is in the shaded region, so it is a solution.
- (−1, 5): This point is not in the shaded region, so it is not a solution.
- (2, −1): This point is not in the shaded region, so it is not a solution.
- (−2, −5): This point is in the shaded region, so it is a solution.
Therefore, the points (0, 0) and (−2, −5) lie in the solution set to the system of inequalities.
To find the points that lie in the solution set to the system of inequalities, we need to determine which points satisfy both inequalities. Let's solve each inequality separately:
1. y < -3x + 3:
To graph this inequality, let's convert it to the slope-intercept form (y = mx + b):
y = -3x + 3.
Now, let's choose a point to test if it satisfies the inequality, such as (0,0):
Substituting the values into the inequality:
0 < -3(0) + 3
0 < 3
Since 0 is less than 3, the point (0, 0) satisfies the inequality.
2. y < x + 2:
Again, let's convert this inequality to the slope-intercept form:
y = x + 2.
Let's pick another point to test, such as (1,1):
Substituting the values into the inequality:
1 < 1 + 2
1 < 3
Since 1 is less than 3, the point (1, 1) satisfies the inequality.
In conclusion, the points (0, 0) and (1, 1) lie in the solution set of the given system of inequalities.