IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.
Sagot :
Let's analyze the system of inequalities provided:
1. [tex]\( y > -3x + 5 \)[/tex]
2. [tex]\( y > x - 2 \)[/tex]
We are given four points, and we need to determine which of these points satisfies both inequalities. Let's check each point one by one:
### Point A: [tex]\((3, -2)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[ -2 > -3(3) + 5 \][/tex]
[tex]\[ -2 > -9 + 5 \][/tex]
[tex]\[ -2 > -4 \][/tex]
This is a true statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[ -2 > 3 - 2 \][/tex]
[tex]\[ -2 > 1 \][/tex]
This is a false statement.
Since Point A does not satisfy both inequalities, it is not a solution.
### Point B: [tex]\((4, 1)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 > -3(4) + 5 \][/tex]
[tex]\[ 1 > -12 + 5 \][/tex]
[tex]\[ 1 > -7 \][/tex]
This is a true statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 > 4 - 2 \][/tex]
[tex]\[ 1 > 2 \][/tex]
This is a false statement.
Since Point B does not satisfy both inequalities, it is not a solution.
### Point C: [tex]\((-1, 4)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 > -3(-1) + 5 \][/tex]
[tex]\[ 4 > 3 + 5 \][/tex]
[tex]\[ 4 > 8 \][/tex]
This is a false statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 > -1 - 2 \][/tex]
[tex]\[ 4 > -3 \][/tex]
This is a true statement.
Since Point C does not satisfy both inequalities, it is not a solution.
### Point D: [tex]\((2, 3)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 > -3(2) + 5 \][/tex]
[tex]\[ 3 > -6 + 5 \][/tex]
[tex]\[ 3 > -1 \][/tex]
This is a true statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 > 2 - 2 \][/tex]
[tex]\[ 3 > 0 \][/tex]
This is also a true statement.
Since Point D satisfies both inequalities, it is the solution.
Therefore, the correct answer is:
D. [tex]\((2, 3)\)[/tex]
1. [tex]\( y > -3x + 5 \)[/tex]
2. [tex]\( y > x - 2 \)[/tex]
We are given four points, and we need to determine which of these points satisfies both inequalities. Let's check each point one by one:
### Point A: [tex]\((3, -2)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[ -2 > -3(3) + 5 \][/tex]
[tex]\[ -2 > -9 + 5 \][/tex]
[tex]\[ -2 > -4 \][/tex]
This is a true statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[ -2 > 3 - 2 \][/tex]
[tex]\[ -2 > 1 \][/tex]
This is a false statement.
Since Point A does not satisfy both inequalities, it is not a solution.
### Point B: [tex]\((4, 1)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 > -3(4) + 5 \][/tex]
[tex]\[ 1 > -12 + 5 \][/tex]
[tex]\[ 1 > -7 \][/tex]
This is a true statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[ 1 > 4 - 2 \][/tex]
[tex]\[ 1 > 2 \][/tex]
This is a false statement.
Since Point B does not satisfy both inequalities, it is not a solution.
### Point C: [tex]\((-1, 4)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 > -3(-1) + 5 \][/tex]
[tex]\[ 4 > 3 + 5 \][/tex]
[tex]\[ 4 > 8 \][/tex]
This is a false statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 > -1 - 2 \][/tex]
[tex]\[ 4 > -3 \][/tex]
This is a true statement.
Since Point C does not satisfy both inequalities, it is not a solution.
### Point D: [tex]\((2, 3)\)[/tex]
1. For the first inequality [tex]\( y > -3x + 5 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 > -3(2) + 5 \][/tex]
[tex]\[ 3 > -6 + 5 \][/tex]
[tex]\[ 3 > -1 \][/tex]
This is a true statement.
2. For the second inequality [tex]\( y > x - 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex]:
[tex]\[ 3 > 2 - 2 \][/tex]
[tex]\[ 3 > 0 \][/tex]
This is also a true statement.
Since Point D satisfies both inequalities, it is the solution.
Therefore, the correct answer is:
D. [tex]\((2, 3)\)[/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.
