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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]
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