To determine which ordered pair satisfies both inequalities:
[tex]\[
\begin{array}{l}
y < 3x - 1 \\
y \geq -x + 4
\end{array}
\][/tex]
we will check each pair individually:
1. Check the pair [tex]\((4, 0)\)[/tex]:
[tex]\[
\begin{array}{l}
\text{First inequality:} \\
0 < 3(4) - 1 \\
0 < 12 - 1 \\
0 < 11 \quad \text{(True)}
\end{array}
\][/tex]
[tex]\[
\begin{array}{l}
\text{Second inequality:} \\
0 \geq -(4) + 4 \\
0 \geq -4 + 4 \\
0 \geq 0 \quad \text{(True)}
\end{array}
\][/tex]
Both inequalities are satisfied by the pair [tex]\((4, 0)\)[/tex].
2. Check the pair [tex]\((1, 2)\)[/tex]:
[tex]\[
\begin{array}{l}
\text{First inequality:} \\
2 < 3(1) - 1 \\
2 < 3 - 1 \\
2 < 2 \quad \text{(False)}
\end{array}
\][/tex]
[tex]\[
\begin{array}{l}
\text{Second inequality:} \\
2 \geq -(1) + 4 \\
2 \geq -1 + 4 \\
2 \geq 3 \quad \text{(False)}
\end{array}
\][/tex]
Neither inequality is satisfied by the pair [tex]\((1, 2)\)[/tex].
3. Check the pair [tex]\((0, 4)\)[/tex]:
[tex]\[
\begin{array}{l}
\text{First inequality:} \\
4 < 3(0) - 1 \\
4 < 0 - 1 \\
4 < -1 \quad \text{(False)}
\end{array}
\][/tex]
[tex]\[
\begin{array}{l}
\text{Second inequality:} \\
4 \geq -(0) + 4 \\
4 \geq 0 + 4 \\
4 \geq 4 \quad \text{(True)}
\end{array}
\][/tex]
Only the second inequality is satisfied by the pair [tex]\((0, 4)\)[/tex].
After checking all pairs, the only pair that satisfies both inequalities is [tex]\((4, 0)\)[/tex].
