To determine which ordered pair [tex]\((x, y)\)[/tex] satisfies the system of inequalities:
[tex]\[
\begin{cases}
y < x^2 + 3 \\
y > x^2 - 2x + 8
\end{cases}
\][/tex]
we need to check each pair against these inequalities step-by-step.
Let's evaluate each pair:
### Pair [tex]\((-4, 2)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[
2 < (-4)^2 + 3 \Rightarrow 2 < 16 + 3 \Rightarrow 2 < 19
\][/tex]
This is true.
2. Second Inequality: [tex]\( y > x^2 - 2x + 8 \)[/tex]
[tex]\[
2 > (-4)^2 - 2(-4) + 8 \Rightarrow 2 > 16 + 8 + 8 \Rightarrow 2 > 32
\][/tex]
This is false.
Since one of the inequalities is not satisfied, [tex]\((-4, 2)\)[/tex] is not a solution.
### Pair [tex]\((0, 6)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[
6 < 0^2 + 3 \Rightarrow 6 < 3
\][/tex]
This is false.
Since the first inequality is not satisfied, [tex]\((0, 6)\)[/tex] is not a solution.
### Pair [tex]\((1, 12)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[
12 < 1^2 + 3 \Rightarrow 12 < 1 + 3 \Rightarrow 12 < 4
\][/tex]
This is false.
Since the first inequality is not satisfied, [tex]\((1, 12)\)[/tex] is not a solution.
### Pair [tex]\((4, 18)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[
18 < 4^2 + 3 \Rightarrow 18 < 16 + 3 \Rightarrow 18 < 19
\][/tex]
This is true.
2. Second Inequality: [tex]\( y > x^2 - 2x + 8 \)[/tex]
[tex]\[
18 > 4^2 - 2(4) + 8 \Rightarrow 18 > 16 - 8 + 8 \Rightarrow 18 > 16
\][/tex]
This is true.
Since both inequalities are satisfied, [tex]\((4, 18)\)[/tex] is a solution.
Thus, the ordered pair that is included in the solution set for the given system of inequalities is [tex]\((4, 18)\)[/tex].
