To determine which of the given ordered pairs satisfies the inequality [tex]\( y > \frac{1}{2} x + 5 \)[/tex], we'll test each pair individually by substituting the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values into the inequality.
1. Test the pair [tex]\((10, 9)\)[/tex]:
[tex]\[
\text{Substitute } x = 10 \text{ and } y = 9 \text{ into the inequality } y > \frac{1}{2} x + 5.
\][/tex]
[tex]\[
9 > \frac{1}{2} \cdot 10 + 5
\][/tex]
[tex]\[
9 > 5 + 5
\][/tex]
[tex]\[
9 > 10 \quad \text{(False)}
\][/tex]
So, the pair [tex]\((10, 9)\)[/tex] does not satisfy the inequality.
2. Test the pair [tex]\((8, 10)\)[/tex]:
[tex]\[
\text{Substitute } x = 8 \text{ and } y = 10 \text{ into the inequality } y > \frac{1}{2} x + 5.
\][/tex]
[tex]\[
10 > \frac{1}{2} \cdot 8 + 5
\][/tex]
[tex]\[
10 > 4 + 5
\][/tex]
[tex]\[
10 > 9 \quad \text{(True)}
\][/tex]
So, the pair [tex]\((8, 10)\)[/tex] satisfies the inequality.
3. Test the pair [tex]\((4, 6)\)[/tex]:
[tex]\[
\text{Substitute } x = 4 \text{ and } y = 6 \text{ into the inequality } y > \frac{1}{2} x + 5.
\][/tex]
[tex]\[
6 > \frac{1}{2} \cdot 4 + 5
\][/tex]
[tex]\[
6 > 2 + 5
\][/tex]
[tex]\[
6 > 7 \quad \text{(False)}
\][/tex]
So, the pair [tex]\((4, 6)\)[/tex] does not satisfy the inequality.
4. Test the pair [tex]\((2, 6)\)[/tex]:
[tex]\[
\text{Substitute } x = 2 \text{ and } y = 6 \text{ into the inequality } y > \frac{1}{2} x + 5.
\][/tex]
[tex]\[
6 > \frac{1}{2} \cdot 2 + 5
\][/tex]
[tex]\[
6 > 1 + 5
\][/tex]
[tex]\[
6 > 6 \quad \text{(False)}
\][/tex]
So, the pair [tex]\((2, 6)\)[/tex] does not satisfy the inequality.
After evaluating each of the ordered pairs, the pair that satisfies the inequality [tex]\( y > \frac{1}{2} x + 5 \)[/tex] is [tex]\((8, 10)\)[/tex].
