To determine which ordered pairs satisfy both of the given inequalities:
[tex]\[
\begin{array}{c}
y > -3x + 4 \\
y \leq 3x - 2
\end{array}
\][/tex]
Let's analyze each ordered pair individually.
1. Ordered pair [tex]\((2,1)\)[/tex]:
- Check the first inequality [tex]\(y > -3x + 4\)[/tex]:
[tex]\[
1 > -3(2) + 4
\][/tex]
[tex]\[
1 > -6 + 4
\][/tex]
[tex]\[
1 > -2 \quad \text{(True)}
\][/tex]
- Check the second inequality [tex]\(y \leq 3x - 2\)[/tex]:
[tex]\[
1 \leq 3(2) - 2
\][/tex]
[tex]\[
1 \leq 6 - 2
\][/tex]
[tex]\[
1 \leq 4 \quad \text{(True)}
\][/tex]
Both conditions are satisfied for [tex]\((2, 1)\)[/tex].
2. Ordered pair [tex]\((1, -2)\)[/tex]:
- Check the first inequality [tex]\(y > -3x + 4\)[/tex]:
[tex]\[
-2 > -3(1) + 4
\][/tex]
[tex]\[
-2 > -3 + 4
\][/tex]
[tex]\[
-2 > 1 \quad \text{(False)}
\][/tex]
Since the first condition is not satisfied, there's no need to check the second condition. [tex]\((1, -2)\)[/tex] is not in the solution set.
3. Ordered pair [tex]\((0, 4)\)[/tex]:
- Check the first inequality [tex]\(y > -3x + 4\)[/tex]:
[tex]\[
4 > -3(0) + 4
\][/tex]
[tex]\[
4 > 0 + 4
\][/tex]
[tex]\[
4 > 4 \quad \text{(False)}
\][/tex]
Since the first condition is not satisfied, there's no need to check the second condition. [tex]\((0, 4)\)[/tex] is not in the solution set.
4. Ordered pair [tex]\((1, 3)\)[/tex]:
- Check the first inequality [tex]\(y > -3x + 4\)[/tex]:
[tex]\[
3 > -3(1) + 4
\][/tex]
[tex]\[
3 > -3 + 4
\][/tex]
[tex]\[
3 > 1 \quad \text{(True)}
\][/tex]
- Check the second inequality [tex]\(y \leq 3x - 2\)[/tex]:
[tex]\[
3 \leq 3(1) - 2
\][/tex]
[tex]\[
3 \leq 3 - 2
\][/tex]
[tex]\[
3 \leq 1 \quad \text{(False)}
\][/tex]
Since the second condition is not satisfied, [tex]\((1, 3)\)[/tex] is not in the solution set.
After analyzing all pairs, the only ordered pair that satisfies both inequalities is [tex]\((2,1)\)[/tex].
Hence, the ordered pair that is in the solution set is [tex]\( (2, 1) \)[/tex].
