Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine which points are solutions to the system of inequalities, we need to check each point against all three inequalities:
1. [tex]\( y < 2x \)[/tex]
2. [tex]\( y < 8 \)[/tex]
3. [tex]\( x > 2 \)[/tex]
Let's evaluate each point one by one:
### Point A: [tex]\((5, 3)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 5 \)[/tex], [tex]\( 5 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 3 \)[/tex], [tex]\( 3 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 5 \)[/tex] and [tex]\( y = 3 \)[/tex], [tex]\( 3 < 2 \cdot 5 \rightarrow 3 < 10 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((5, 3)\)[/tex] is a solution.
### Point B: [tex]\((1, -4)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 1 \)[/tex], [tex]\( 1 > 2 \)[/tex] is false.
Since the first inequality is not satisfied, [tex]\((1, -4)\)[/tex] is not a solution.
### Point C: [tex]\((4, -2)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 4 \)[/tex], [tex]\( 4 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = -2 \)[/tex], [tex]\( -2 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 4 \)[/tex] and [tex]\( y = -2 \)[/tex], [tex]\( -2 < 2 \cdot 4 \rightarrow -2 < 8 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((4, -2)\)[/tex] is a solution.
### Point D: [tex]\((3, 3)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 3 \)[/tex], [tex]\( 3 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 3 \)[/tex], [tex]\( 3 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 3 \)[/tex] and [tex]\( y = 3 \)[/tex], [tex]\( 3 < 2 \cdot 3 \rightarrow 3 < 6 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((3, 3)\)[/tex] is a solution.
### Point E: [tex]\((5, 9)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 5 \)[/tex], [tex]\( 5 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 9 \)[/tex], [tex]\( 9 < 8 \)[/tex] is false.
Since the first inequality is not satisfied, [tex]\((5, 9)\)[/tex] is not a solution.
### Point F: [tex]\((6, 0)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 6 \)[/tex], [tex]\( 6 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 0 \)[/tex], [tex]\( 0 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 6 \)[/tex] and [tex]\( y = 0 \)[/tex], [tex]\( 0 < 2 \cdot 6 \rightarrow 0 < 12 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((6, 0)\)[/tex] is a solution.
### Conclusion
The points that are solutions to the system of inequalities are:
- A. [tex]\((5, 3)\)[/tex]
- C. [tex]\((4, -2)\)[/tex]
- D. [tex]\((3, 3)\)[/tex]
- F. [tex]\((6, 0)\)[/tex]
Thus, the correct answers are A, C, D, and F.
1. [tex]\( y < 2x \)[/tex]
2. [tex]\( y < 8 \)[/tex]
3. [tex]\( x > 2 \)[/tex]
Let's evaluate each point one by one:
### Point A: [tex]\((5, 3)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 5 \)[/tex], [tex]\( 5 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 3 \)[/tex], [tex]\( 3 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 5 \)[/tex] and [tex]\( y = 3 \)[/tex], [tex]\( 3 < 2 \cdot 5 \rightarrow 3 < 10 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((5, 3)\)[/tex] is a solution.
### Point B: [tex]\((1, -4)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 1 \)[/tex], [tex]\( 1 > 2 \)[/tex] is false.
Since the first inequality is not satisfied, [tex]\((1, -4)\)[/tex] is not a solution.
### Point C: [tex]\((4, -2)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 4 \)[/tex], [tex]\( 4 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = -2 \)[/tex], [tex]\( -2 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 4 \)[/tex] and [tex]\( y = -2 \)[/tex], [tex]\( -2 < 2 \cdot 4 \rightarrow -2 < 8 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((4, -2)\)[/tex] is a solution.
### Point D: [tex]\((3, 3)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 3 \)[/tex], [tex]\( 3 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 3 \)[/tex], [tex]\( 3 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 3 \)[/tex] and [tex]\( y = 3 \)[/tex], [tex]\( 3 < 2 \cdot 3 \rightarrow 3 < 6 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((3, 3)\)[/tex] is a solution.
### Point E: [tex]\((5, 9)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 5 \)[/tex], [tex]\( 5 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 9 \)[/tex], [tex]\( 9 < 8 \)[/tex] is false.
Since the first inequality is not satisfied, [tex]\((5, 9)\)[/tex] is not a solution.
### Point F: [tex]\((6, 0)\)[/tex]
- Inequality [tex]\( x > 2 \)[/tex]: For [tex]\( x = 6 \)[/tex], [tex]\( 6 > 2 \)[/tex] is true.
- Inequality [tex]\( y < 8 \)[/tex]: For [tex]\( y = 0 \)[/tex], [tex]\( 0 < 8 \)[/tex] is true.
- Inequality [tex]\( y < 2x \)[/tex]: For [tex]\( x = 6 \)[/tex] and [tex]\( y = 0 \)[/tex], [tex]\( 0 < 2 \cdot 6 \rightarrow 0 < 12 \)[/tex] is true.
Since all three inequalities are satisfied, [tex]\((6, 0)\)[/tex] is a solution.
### Conclusion
The points that are solutions to the system of inequalities are:
- A. [tex]\((5, 3)\)[/tex]
- C. [tex]\((4, -2)\)[/tex]
- D. [tex]\((3, 3)\)[/tex]
- F. [tex]\((6, 0)\)[/tex]
Thus, the correct answers are A, C, D, and F.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
