Let's determine which of the given ordered pairs satisfy the system of linear inequalities:
[tex]\[
\begin{cases}
y \geq -\frac{1}{3} x + 2 \\
y < 2x + 3
\end{cases}
\][/tex]
We will check each pair against the inequalities.
### Evaluating the Pairs from the First List [tex]\((2,2), (3,1), (4,2)\)[/tex]:
1. Pair [tex]\((2, 2)\)[/tex]:
- For the first inequality [tex]\( y \geq -\frac{1}{3} x + 2 \)[/tex]:
[tex]\[
2 \geq -\frac{1}{3}(2) + 2 \implies 2 \geq -\frac{2}{3} + 2 \implies 2 \geq \frac{4}{3}
\][/tex]
This is true.
- For the second inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
2 < 2(2) + 3 \implies 2 < 4 + 3 \implies 2 < 7
\][/tex]
This is also true.
Thus, [tex]\((2, 2)\)[/tex] satisfies both inequalities.
2. Pair [tex]\((3, 1)\)[/tex]:
- For the first inequality [tex]\( y \geq -\frac{1}{3} x + 2 \)[/tex]:
[tex]\[
1 \geq -\frac{1}{3}(3) + 2 \implies 1 \geq -1 + 2 \implies 1 \geq 1
\][/tex]
This is true.
- For the second inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
1 < 2(3) + 3 \implies 1 < 6 + 3 \implies 1 < 9
\][/tex]
This is also true.
Thus, [tex]\((3, 1)\)[/tex] satisfies both inequalities.
3. Pair [tex]\((4, 2)\)[/tex]:
- For the first inequality [tex]\( y \geq -\frac{1}{3} x + 2 \)[/tex]:
[tex]\[
2 \geq -\frac{1}{3}(4) + 2 \implies 2 \geq -\frac{4}{3} + 2 \implies 2 \geq \frac{2}{3}
\][/tex]
This is true.
- For the second inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
2 < 2(4) + 3 \implies 2 < 8 + 3 \implies 2 < 11
\][/tex]
This is also true.
Thus, [tex]\((4, 2)\)[/tex] satisfies both inequalities.
So, the solution set from the first list [tex]\((2, 2), (3, 1), (4, 2)\)[/tex] includes all three pairs.
### Evaluating the Pairs from the Second List [tex]\((2,2), (3,-1), (4,1)\)[/tex]:
1. Pair [tex]\((2, 2)\)[/tex]:
- This pair was previously evaluated and found to satisfy both inequalities.
2. Pair [tex]\((3, -1)\)[/tex]:
- For the first inequality [tex]\( y \geq -\frac{1}{3} x + 2 \)[/tex]:
[tex]\[
-1 \geq -\frac{1}{3}(3) + 2 \implies -1 \geq -1 + 2 \implies -1 \geq 1
\][/tex]
This is false.
Thus, [tex]\((3, -1)\)[/tex] does not satisfy the system of inequalities.
3. Pair [tex]\((4, 1)\)[/tex]:
- For the first inequality [tex]\( y \geq -\frac{1}{3} x + 2 \)[/tex]:
[tex]\[
1 \geq -\frac{1}{3}(4) + 2 \implies 1 \geq -\frac{4}{3} + 2 \implies 1 \geq \frac{2}{3}
\][/tex]
This is true.
- For the second inequality [tex]\( y < 2x + 3 \)[/tex]:
[tex]\[
1 < 2(4) + 3 \implies 1 < 8 + 3 \implies 1 < 11
\][/tex]
This is also true.
Thus, [tex]\((4, 1)\)[/tex] satisfies both inequalities.
So, the solution set from the second list [tex]\((2, 2), (3, -1), (4, 1)\)[/tex] includes the pairs [tex]\((2, 2)\)[/tex] and [tex]\((4, 1)\)[/tex].
### Conclusion
The pairs in the solution sets of the two supplied lists are:
1. From the first list [tex]\((2, 2), (3, 1), (4, 2)\)[/tex], the pairs that satisfy the system are:
[tex]\[
\{(2, 2), (3, 1), (4, 2)\}
\][/tex]
2. From the second list [tex]\((2, 2), (3, -1), (4, 1)\)[/tex], the pairs that satisfy the system are:
[tex]\[
\{(2, 2), (4, 1)\}
\][/tex]
