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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.
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