To determine which point is a solution to the given system of inequalities:
[tex]\[
\begin{aligned}
x + 4y & > 12 \\
3y & > x + 6
\end{aligned}
\][/tex]
we will check each point to see if it satisfies both inequalities.
1. Point [tex]\((0, 3)\)[/tex]:
First inequality: [tex]\(x + 4y > 12\)[/tex]
[tex]\[
0 + 4 \cdot 3 > 12 \implies 12 > 12
\][/tex]
This is false because 12 is not greater than 12.
Second inequality: [tex]\(3y > x + 6\)[/tex]
[tex]\[
3 \cdot 3 > 0 + 6 \implies 9 > 6
\][/tex]
This is true.
Since the first inequality is not satisfied by the point [tex]\((0, 3)\)[/tex], it is not a solution to the system.
2. Point [tex]\((-4, 6)\)[/tex]:
First inequality: [tex]\(x + 4y > 12\)[/tex]
[tex]\[
-4 + 4 \cdot 6 > 12 \implies -4 + 24 > 12 \implies 20 > 12
\][/tex]
This is true.
Second inequality: [tex]\(3y > x + 6\)[/tex]
[tex]\[
3 \cdot 6 > -4 + 6 \implies 18 > 2
\][/tex]
This is also true.
Since both inequalities are satisfied by the point [tex]\((-4, 6)\)[/tex], it is a solution to the system.
3. Point [tex]\((5, 2)\)[/tex]:
First inequality: [tex]\(x + 4y > 12\)[/tex]
[tex]\[
5 + 4 \cdot 2 > 12 \implies 5 + 8 > 12 \implies 13 > 12
\][/tex]
This is true.
Second inequality: [tex]\(3y > x + 6\)[/tex]
[tex]\[
3 \cdot 2 > 5 + 6 \implies 6 > 11
\][/tex]
This is false.
Since the second inequality is not satisfied by the point [tex]\((5, 2)\)[/tex], it is not a solution to the system.
4. Point [tex]\((-2, 3)\)[/tex]:
First inequality: [tex]\(x + 4y > 12\)[/tex]
[tex]\[
-2 + 4 \cdot 3 > 12 \implies -2 + 12 > 12 \implies 10 > 12
\][/tex]
This is false.
Second inequality: [tex]\(3y > x + 6\)[/tex]
[tex]\[
3 \cdot 3 > -2 + 6 \implies 9 > 4
\][/tex]
This is true.
Since the first inequality is not satisfied by the point [tex]\((-2, 3)\)[/tex], it is not a solution to the system.
Based on the evaluation of all points, the only point that satisfies both inequalities is:
[tex]\[
\boxed{(-4, 6)}
\][/tex]
