Answer :
To determine which ordered pair makes both inequalities true, we need to test the given pairs against each inequality.
The inequalities are:
[tex]\[ 1. \quad y < 3x - 1 \][/tex]
[tex]\[ 2. \quad y \geq -x + 4 \][/tex]
We will test each ordered pair [tex]\((x, y)\)[/tex].
### Testing the pair [tex]\((4, 0)\)[/tex]:
For the first inequality:
[tex]\[ y < 3x - 1 \implies 0 < 3(4) - 1 \implies 0 < 12 - 1 \implies 0 < 11 \quad (\text{True}) \][/tex]
For the second inequality:
[tex]\[ y \geq -x + 4 \implies 0 \geq -4 + 4 \implies 0 \geq 0 \quad (\text{True}) \][/tex]
Since both inequalities are true for [tex]\((4, 0)\)[/tex], this pair satisfies both inequalities.
### Testing the pair [tex]\((1, 2)\)[/tex]:
For the first inequality:
[tex]\[ y < 3x - 1 \implies 2 < 3(1) - 1 \implies 2 < 3 - 1 \implies 2 < 2 \quad (\text{False}) \][/tex]
Since the first inequality is false, we do not need to check the second inequality for this pair. Therefore, [tex]\((1, 2)\)[/tex] does not satisfy both inequalities.
### Testing the pair [tex]\((0, 4)\)[/tex]:
For the first inequality:
[tex]\[ y < 3x - 1 \implies 4 < 3(0) - 1 \implies 4 < 0 - 1 \implies 4 < -1 \quad (\text{False}) \][/tex]
Since the first inequality is false, we do not need to check the second inequality for this pair. Therefore, [tex]\((0, 4)\)[/tex] does not satisfy both inequalities.
### Conclusion:
The only ordered pair that makes both inequalities true is [tex]\((4, 0)\)[/tex].
The inequalities are:
[tex]\[ 1. \quad y < 3x - 1 \][/tex]
[tex]\[ 2. \quad y \geq -x + 4 \][/tex]
We will test each ordered pair [tex]\((x, y)\)[/tex].
### Testing the pair [tex]\((4, 0)\)[/tex]:
For the first inequality:
[tex]\[ y < 3x - 1 \implies 0 < 3(4) - 1 \implies 0 < 12 - 1 \implies 0 < 11 \quad (\text{True}) \][/tex]
For the second inequality:
[tex]\[ y \geq -x + 4 \implies 0 \geq -4 + 4 \implies 0 \geq 0 \quad (\text{True}) \][/tex]
Since both inequalities are true for [tex]\((4, 0)\)[/tex], this pair satisfies both inequalities.
### Testing the pair [tex]\((1, 2)\)[/tex]:
For the first inequality:
[tex]\[ y < 3x - 1 \implies 2 < 3(1) - 1 \implies 2 < 3 - 1 \implies 2 < 2 \quad (\text{False}) \][/tex]
Since the first inequality is false, we do not need to check the second inequality for this pair. Therefore, [tex]\((1, 2)\)[/tex] does not satisfy both inequalities.
### Testing the pair [tex]\((0, 4)\)[/tex]:
For the first inequality:
[tex]\[ y < 3x - 1 \implies 4 < 3(0) - 1 \implies 4 < 0 - 1 \implies 4 < -1 \quad (\text{False}) \][/tex]
Since the first inequality is false, we do not need to check the second inequality for this pair. Therefore, [tex]\((0, 4)\)[/tex] does not satisfy both inequalities.
### Conclusion:
The only ordered pair that makes both inequalities true is [tex]\((4, 0)\)[/tex].
