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Sagot :
Certainly! To determine which points satisfy the system of inequalities [tex]\( y > 5x - 2 \)[/tex] and [tex]\( y < 5x + 3 \)[/tex], we will check each point one by one.
### Point A: [tex]\((4, 23)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 23 > 5(4) - 2 \rightarrow 23 > 20 - 2 \rightarrow 23 > 18 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 23 < 5(4) + 3 \rightarrow 23 < 20 + 3 \rightarrow 23 < 23 \][/tex]
This inequality is false.
Therefore, point [tex]\((4, 23)\)[/tex] does not satisfy both inequalities.
### Point B: [tex]\((-5, 25)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 25 > 5(-5) - 2 \rightarrow 25 > -25 - 2 \rightarrow 25 > -27 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 25 < 5(-5) + 3 \rightarrow 25 < -25 + 3 \rightarrow 25 < -22 \][/tex]
This inequality is false.
Therefore, point [tex]\((-5, 25)\)[/tex] does not satisfy both inequalities.
### Point C: [tex]\((5, 28)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 28 > 5(5) - 2 \rightarrow 28 > 25 - 2 \rightarrow 28 > 23 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 28 < 5(5) + 3 \rightarrow 28 < 25 + 3 \rightarrow 28 < 28 \][/tex]
This inequality is false.
Therefore, point [tex]\((5, 28)\)[/tex] does not satisfy both inequalities.
### Point D: [tex]\((4, 20)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 20 > 5(4) - 2 \rightarrow 20 > 20 - 2 \rightarrow 20 > 18 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 20 < 5(4) + 3 \rightarrow 20 < 20 + 3 \rightarrow 20 < 23 \][/tex]
This inequality is true.
Therefore, point [tex]\((4, 20)\)[/tex] satisfies both inequalities.
### Conclusion
The point that satisfies both inequalities is:
- [tex]\( (4, 20) \)[/tex]
Hence, the answer is:
D. [tex]\((4, 20)\)[/tex]
### Point A: [tex]\((4, 23)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 23 > 5(4) - 2 \rightarrow 23 > 20 - 2 \rightarrow 23 > 18 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 23 < 5(4) + 3 \rightarrow 23 < 20 + 3 \rightarrow 23 < 23 \][/tex]
This inequality is false.
Therefore, point [tex]\((4, 23)\)[/tex] does not satisfy both inequalities.
### Point B: [tex]\((-5, 25)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 25 > 5(-5) - 2 \rightarrow 25 > -25 - 2 \rightarrow 25 > -27 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 25 < 5(-5) + 3 \rightarrow 25 < -25 + 3 \rightarrow 25 < -22 \][/tex]
This inequality is false.
Therefore, point [tex]\((-5, 25)\)[/tex] does not satisfy both inequalities.
### Point C: [tex]\((5, 28)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 28 > 5(5) - 2 \rightarrow 28 > 25 - 2 \rightarrow 28 > 23 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 28 < 5(5) + 3 \rightarrow 28 < 25 + 3 \rightarrow 28 < 28 \][/tex]
This inequality is false.
Therefore, point [tex]\((5, 28)\)[/tex] does not satisfy both inequalities.
### Point D: [tex]\((4, 20)\)[/tex]
1. Check [tex]\( y > 5x - 2 \)[/tex]:
[tex]\[ 20 > 5(4) - 2 \rightarrow 20 > 20 - 2 \rightarrow 20 > 18 \][/tex]
This inequality is true.
2. Check [tex]\( y < 5x + 3 \)[/tex]:
[tex]\[ 20 < 5(4) + 3 \rightarrow 20 < 20 + 3 \rightarrow 20 < 23 \][/tex]
This inequality is true.
Therefore, point [tex]\((4, 20)\)[/tex] satisfies both inequalities.
### Conclusion
The point that satisfies both inequalities is:
- [tex]\( (4, 20) \)[/tex]
Hence, the answer is:
D. [tex]\((4, 20)\)[/tex]
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