To determine which ordered pair is included in the solution set to the given system of inequalities, we will evaluate each pair against the inequalities provided. The system of inequalities is:
1. [tex]\( y < x^2 + 3 \)[/tex]
2. [tex]\( y > x^2 - 2x + 8 \)[/tex]
Let's check each given pair [tex]\((x, y)\)[/tex] one by one.
### For the pair [tex]\((-4, 2)\)[/tex]:
1. Inequality check for [tex]\( y < x^2 + 3 \)[/tex]:
[tex]\[
2 < (-4)^2 + 3 \implies 2 < 16 + 3 \implies 2 < 19 \quad \text{(True)}
\][/tex]
2. Inequality check for [tex]\( y > x^2 - 2x + 8 \)[/tex]:
[tex]\[
2 > (-4)^2 - 2(-4) + 8 \implies 2 > 16 + 8 + 8 \implies 2 > 32 \quad \text{(False)}
\][/tex]
Since the second inequality is false, the pair [tex]\((-4, 2)\)[/tex] is not included in the solution set.
### For the pair [tex]\((0, 6)\)[/tex]:
1. Inequality check for [tex]\( y < x^2 + 3 \)[/tex]:
[tex]\[
6 < 0^2 + 3 \implies 6 < 3 \quad \text{(False)}
\][/tex]
Since the first inequality is false, the pair [tex]\((0, 6)\)[/tex] is not included in the solution set.
### For the pair [tex]\((1, 12)\)[/tex]:
1. Inequality check for [tex]\( y < x^2 + 3 \)[/tex]:
[tex]\[
12 < 1^2 + 3 \implies 12 < 1 + 3 \implies 12 < 4 \quad \text{(False)}
\][/tex]
Since the first inequality is false, the pair [tex]\((1, 12)\)[/tex] is not included in the solution set.
### For the pair [tex]\((4, 18)\)[/tex]:
1. Inequality check for [tex]\( y < x^2 + 3 \)[/tex]:
[tex]\[
18 < 4^2 + 3 \implies 18 < 16 + 3 \implies 18 < 19 \quad \text{(True)}
\][/tex]
2. Inequality check for [tex]\( y > x^2 - 2x + 8 \)[/tex]:
[tex]\[
18 > 4^2 - 2(4) + 8 \implies 18 > 16 - 8 + 8 \implies 18 > 16 \quad \text{(True)}
\][/tex]
Since both inequalities are true, the pair [tex]\((4, 18)\)[/tex] is included in the solution set.
### Conclusion:
The ordered pair that is included in the solution set to the given system of inequalities is:
[tex]\[
(4, 18)
\][/tex]
