Answer :
To determine which ordered pair makes both inequalities true, we need to check each pair against the two given inequalities:
1. [tex]\( y < 3x - 1 \)[/tex]
2. [tex]\( y \geq -x + 4 \)[/tex]
Let’s evaluate each pair one by one.
### For the pair [tex]\((4, 0)\)[/tex]:
1. Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 0\)[/tex] into the first inequality:
[tex]\[ 0 < 3(4) - 1 \][/tex]
[tex]\[ 0 < 12 - 1 \][/tex]
[tex]\[ 0 < 11 \][/tex]
This inequality is true.
2. Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 0\)[/tex] into the second inequality:
[tex]\[ 0 \geq -4 + 4 \][/tex]
[tex]\[ 0 \geq 0 \][/tex]
This inequality is true as well.
Since both inequalities are satisfied, [tex]\((4, 0)\)[/tex] makes both inequalities true.
### For the pair [tex]\((1, 2)\)[/tex]:
1. Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 2\)[/tex] into the first inequality:
[tex]\[ 2 < 3(1) - 1 \][/tex]
[tex]\[ 2 < 3 - 1 \][/tex]
[tex]\[ 2 < 2 \][/tex]
This inequality is false.
Since the first inequality is not satisfied, we don't need to check the second inequality for this pair. [tex]\((1, 2)\)[/tex] does not make both inequalities true.
### For the pair [tex]\((0, 4)\)[/tex]:
1. Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 4\)[/tex] into the first inequality:
[tex]\[ 4 < 3(0) - 1 \][/tex]
[tex]\[ 4 < -1 \][/tex]
This inequality is false.
Since the first inequality is not satisfied, we don't need to check the second inequality for this pair. [tex]\((0, 4)\)[/tex] does not make both inequalities true.
### For the pair [tex]\((2, 1)\)[/tex]:
1. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into the first inequality:
[tex]\[ 1 < 3(2) - 1 \][/tex]
[tex]\[ 1 < 6 - 1 \][/tex]
[tex]\[ 1 < 5 \][/tex]
This inequality is true.
2. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into the second inequality:
[tex]\[ 1 \geq -2 + 4 \][/tex]
[tex]\[ 1 \geq 2 \][/tex]
This inequality is false.
Since the second inequality is not satisfied, [tex]\((2, 1)\)[/tex] does not make both inequalities true.
Based on the evaluation, the ordered pair that makes both inequalities true is [tex]\((4, 0)\)[/tex].
1. [tex]\( y < 3x - 1 \)[/tex]
2. [tex]\( y \geq -x + 4 \)[/tex]
Let’s evaluate each pair one by one.
### For the pair [tex]\((4, 0)\)[/tex]:
1. Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 0\)[/tex] into the first inequality:
[tex]\[ 0 < 3(4) - 1 \][/tex]
[tex]\[ 0 < 12 - 1 \][/tex]
[tex]\[ 0 < 11 \][/tex]
This inequality is true.
2. Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 0\)[/tex] into the second inequality:
[tex]\[ 0 \geq -4 + 4 \][/tex]
[tex]\[ 0 \geq 0 \][/tex]
This inequality is true as well.
Since both inequalities are satisfied, [tex]\((4, 0)\)[/tex] makes both inequalities true.
### For the pair [tex]\((1, 2)\)[/tex]:
1. Substitute [tex]\(x = 1\)[/tex] and [tex]\(y = 2\)[/tex] into the first inequality:
[tex]\[ 2 < 3(1) - 1 \][/tex]
[tex]\[ 2 < 3 - 1 \][/tex]
[tex]\[ 2 < 2 \][/tex]
This inequality is false.
Since the first inequality is not satisfied, we don't need to check the second inequality for this pair. [tex]\((1, 2)\)[/tex] does not make both inequalities true.
### For the pair [tex]\((0, 4)\)[/tex]:
1. Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 4\)[/tex] into the first inequality:
[tex]\[ 4 < 3(0) - 1 \][/tex]
[tex]\[ 4 < -1 \][/tex]
This inequality is false.
Since the first inequality is not satisfied, we don't need to check the second inequality for this pair. [tex]\((0, 4)\)[/tex] does not make both inequalities true.
### For the pair [tex]\((2, 1)\)[/tex]:
1. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into the first inequality:
[tex]\[ 1 < 3(2) - 1 \][/tex]
[tex]\[ 1 < 6 - 1 \][/tex]
[tex]\[ 1 < 5 \][/tex]
This inequality is true.
2. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into the second inequality:
[tex]\[ 1 \geq -2 + 4 \][/tex]
[tex]\[ 1 \geq 2 \][/tex]
This inequality is false.
Since the second inequality is not satisfied, [tex]\((2, 1)\)[/tex] does not make both inequalities true.
Based on the evaluation, the ordered pair that makes both inequalities true is [tex]\((4, 0)\)[/tex].
