IDNLearn.com provides a comprehensive solution for all your question and answer needs. Discover reliable answers to your questions with our extensive database of expert knowledge.
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
