Which of the following points lie in the solution set to the following system of inequalities?

y < −3x + 3
y < x + 2

(1, −5)
(1, 5)
(5, 1)
(−1, 5)

try them all. I see the first set as the only one allowed.

To determine which of the given points lie in the solution set to the system of inequalities, we need to substitute the x and y coordinates of each point into the inequalities and see if they satisfy the given system.

Let's go through each point one by one:

1. Point (1, -5)
Substitute x = 1 and y = -5 into the inequalities:
-5 < -3(1) + 3
-5 < -3 + 3
-5 < 0

Since -5 is not less than 0, this point does not satisfy the first inequality.
Now, let's check the second inequality:
-5 < 1 + 2
-5 < 3

Since -5 is less than 3, this point satisfies the second inequality.

2. Point (1, 5)
Substitute x = 1 and y = 5 into the inequalities:
5 < -3(1) + 3
5 < -3 + 3
5 < 0

Again, 5 is not less than 0, so this point does not satisfy the first inequality.
Now, check the second inequality:
5 < 1 + 2
5 < 3

As before, 5 is not less than 3, so this point does not satisfy the second inequality.

3. Point (5, 1)
Substitute x = 5 and y = 1 into the inequalities:
1 < -3(5) + 3
1 < -15 + 3
1 < -12

Since 1 is not less than -12, this point does not satisfy the first inequality.
Now, check the second inequality:
1 < 5 + 2
1 < 7

This time, 1 is indeed less than 7, so this point satisfies the second inequality.

4. Point (-1, 5)
Substitute x = -1 and y = 5 into the inequalities:
5 < -3(-1) + 3
5 < 3 + 3
5 < 6

Lastly, 5 is not less than 6, so this point does not satisfy the first inequality.
Now, check the second inequality:
5 < -1 + 2
5 < 1

Again, 5 is not less than 1, so this point does not satisfy the second inequality.

Based on our analysis, only the point (1, -5) lies in the solution set to the given system of inequalities.

To determine which points lie in the solution set to the system of inequalities, we can check if each point satisfies both inequalities.

Let's evaluate each point:
Point (1, -5):
- For the first inequality, substitute x = 1 and y = -5:
-5 < -3(1) + 3
-5 < -3 + 3
-5 < 0
This point satisfies the first inequality.
- For the second inequality, substitute x = 1 and y = -5:
-5 < 1 + 2
-5 < 3
This point also satisfies the second inequality.

Point (1, 5):
- For the first inequality, substitute x = 1 and y = 5:
5 < -3(1) + 3
5 < -3 + 3
5 < 0
This point does not satisfy the first inequality.
- Since it fails to satisfy the first inequality, we don't need to check the second inequality.

Point (5, 1):
- For the first inequality, substitute x = 5 and y = 1:
1 < -3(5) + 3
1 < -15 + 3
1 < -12
This point satisfies the first inequality.
- For the second inequality, substitute x = 5 and y = 1:
1 < 5 + 2
1 < 7
This point also satisfies the second inequality.

Point (-1, 5):
- For the first inequality, substitute x = -1 and y = 5:
5 < -3(-1) + 3
5 < 3 + 3
5 < 6
This point satisfies the first inequality.
- For the second inequality, substitute x = -1 and y = 5:
5 < -1 + 2
5 < 1
This point also satisfies the second inequality.

From our evaluation:
- Point (1, -5) is in the solution set.
- Point (1, 5) is not in the solution set.
- Point (5, 1) is in the solution set.
- Point (-1, 5) is in the solution set.

Therefore, the points that lie in the solution set to the system of inequalities are (1, -5), (5, 1), and (-1, 5).