At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine which ordered pair [tex]\((x, y)\)[/tex] satisfies the system of inequalities:
[tex]\[ \begin{cases} y < x^2 + 3 \\ y > x^2 - 2x + 8 \end{cases} \][/tex]
we need to check each pair against these inequalities step-by-step.
Let's evaluate each pair:
### Pair [tex]\((-4, 2)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 2 < (-4)^2 + 3 \Rightarrow 2 < 16 + 3 \Rightarrow 2 < 19 \][/tex]
This is true.
2. Second Inequality: [tex]\( y > x^2 - 2x + 8 \)[/tex]
[tex]\[ 2 > (-4)^2 - 2(-4) + 8 \Rightarrow 2 > 16 + 8 + 8 \Rightarrow 2 > 32 \][/tex]
This is false.
Since one of the inequalities is not satisfied, [tex]\((-4, 2)\)[/tex] is not a solution.
### Pair [tex]\((0, 6)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 6 < 0^2 + 3 \Rightarrow 6 < 3 \][/tex]
This is false.
Since the first inequality is not satisfied, [tex]\((0, 6)\)[/tex] is not a solution.
### Pair [tex]\((1, 12)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 12 < 1^2 + 3 \Rightarrow 12 < 1 + 3 \Rightarrow 12 < 4 \][/tex]
This is false.
Since the first inequality is not satisfied, [tex]\((1, 12)\)[/tex] is not a solution.
### Pair [tex]\((4, 18)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 18 < 4^2 + 3 \Rightarrow 18 < 16 + 3 \Rightarrow 18 < 19 \][/tex]
This is true.
2. Second Inequality: [tex]\( y > x^2 - 2x + 8 \)[/tex]
[tex]\[ 18 > 4^2 - 2(4) + 8 \Rightarrow 18 > 16 - 8 + 8 \Rightarrow 18 > 16 \][/tex]
This is true.
Since both inequalities are satisfied, [tex]\((4, 18)\)[/tex] is a solution.
Thus, the ordered pair that is included in the solution set for the given system of inequalities is [tex]\((4, 18)\)[/tex].
[tex]\[ \begin{cases} y < x^2 + 3 \\ y > x^2 - 2x + 8 \end{cases} \][/tex]
we need to check each pair against these inequalities step-by-step.
Let's evaluate each pair:
### Pair [tex]\((-4, 2)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 2 < (-4)^2 + 3 \Rightarrow 2 < 16 + 3 \Rightarrow 2 < 19 \][/tex]
This is true.
2. Second Inequality: [tex]\( y > x^2 - 2x + 8 \)[/tex]
[tex]\[ 2 > (-4)^2 - 2(-4) + 8 \Rightarrow 2 > 16 + 8 + 8 \Rightarrow 2 > 32 \][/tex]
This is false.
Since one of the inequalities is not satisfied, [tex]\((-4, 2)\)[/tex] is not a solution.
### Pair [tex]\((0, 6)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 6 < 0^2 + 3 \Rightarrow 6 < 3 \][/tex]
This is false.
Since the first inequality is not satisfied, [tex]\((0, 6)\)[/tex] is not a solution.
### Pair [tex]\((1, 12)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 12 < 1^2 + 3 \Rightarrow 12 < 1 + 3 \Rightarrow 12 < 4 \][/tex]
This is false.
Since the first inequality is not satisfied, [tex]\((1, 12)\)[/tex] is not a solution.
### Pair [tex]\((4, 18)\)[/tex]
1. First Inequality: [tex]\( y < x^2 + 3 \)[/tex]
[tex]\[ 18 < 4^2 + 3 \Rightarrow 18 < 16 + 3 \Rightarrow 18 < 19 \][/tex]
This is true.
2. Second Inequality: [tex]\( y > x^2 - 2x + 8 \)[/tex]
[tex]\[ 18 > 4^2 - 2(4) + 8 \Rightarrow 18 > 16 - 8 + 8 \Rightarrow 18 > 16 \][/tex]
This is true.
Since both inequalities are satisfied, [tex]\((4, 18)\)[/tex] is a solution.
Thus, the ordered pair that is included in the solution set for the given system of inequalities is [tex]\((4, 18)\)[/tex].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.