Join the IDNLearn.com community and get your questions answered by experts. Whether it's a simple query or a complex problem, our experts have the answers you need.
Sagot :
Let's analyze the given inequalities and the sets of points step-by-step to determine which set satisfies both inequalities. The inequalities are:
[tex]\[ y \geq -\frac{1}{3} x + 2 \][/tex]
[tex]\[ y < 2x + 3 \][/tex]
We have four sets of points to check:
1. [tex]\((2,2), (3,1), (4,2)\)[/tex]
2. [tex]\((2,2), (3,-1), (4,1)\)[/tex]
3. [tex]\((2,2), (1,-2), (0,2)\)[/tex]
4. [tex]\((2,2), (1,2), (2,0)\)[/tex]
### Set 1: [tex]\((2,2), (3,1), (4,2)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((3, 1)\)[/tex]:
- [tex]\[ y = 1 \geq -\frac{1}{3}(3) + 2 = 1 \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 1 < 2(3) + 3 = 9 \rightarrow (\text{True}) \][/tex]
3. For the point [tex]\((4, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(4) + 2 = \frac{2}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(4) + 3 = 11 \rightarrow (\text{True}) \][/tex]
All three points satisfy both inequalities. Therefore, Set 1 is valid.
### Set 2: [tex]\((2,2), (3,-1), (4,1)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((3, -1)\)[/tex]:
- [tex]\[ y = -1 \geq -\frac{1}{3}(3) + 2 = 1 \rightarrow (\text{False}) \][/tex]
Since the point [tex]\((3, -1)\)[/tex] does not satisfy the first inequality, Set 2 is invalid.
### Set 3: [tex]\((2,2), (1,-2), (0,2)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((1, -2)\)[/tex]:
- [tex]\[ y = -2 \geq -\frac{1}{3}(1) + 2 = \frac{5}{3} \rightarrow (\text{False}) \][/tex]
Since the point [tex]\((1, -2)\)[/tex] does not satisfy the first inequality, Set 3 is invalid.
### Set 4: [tex]\((2,2), (1,2), (2,0)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((1, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(1) + 2 = \frac{7}{3} \rightarrow (\text{False}) \][/tex]
Since the point [tex]\((1, 2)\)[/tex] does not satisfy the first inequality, Set 4 is invalid.
### Conclusion
Only Set 1: [tex]\((2,2), (3,1), (4,2)\)[/tex] satisfies both inequalities. Thus, the valid set of points is Set 1.
[tex]\[ y \geq -\frac{1}{3} x + 2 \][/tex]
[tex]\[ y < 2x + 3 \][/tex]
We have four sets of points to check:
1. [tex]\((2,2), (3,1), (4,2)\)[/tex]
2. [tex]\((2,2), (3,-1), (4,1)\)[/tex]
3. [tex]\((2,2), (1,-2), (0,2)\)[/tex]
4. [tex]\((2,2), (1,2), (2,0)\)[/tex]
### Set 1: [tex]\((2,2), (3,1), (4,2)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((3, 1)\)[/tex]:
- [tex]\[ y = 1 \geq -\frac{1}{3}(3) + 2 = 1 \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 1 < 2(3) + 3 = 9 \rightarrow (\text{True}) \][/tex]
3. For the point [tex]\((4, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(4) + 2 = \frac{2}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(4) + 3 = 11 \rightarrow (\text{True}) \][/tex]
All three points satisfy both inequalities. Therefore, Set 1 is valid.
### Set 2: [tex]\((2,2), (3,-1), (4,1)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((3, -1)\)[/tex]:
- [tex]\[ y = -1 \geq -\frac{1}{3}(3) + 2 = 1 \rightarrow (\text{False}) \][/tex]
Since the point [tex]\((3, -1)\)[/tex] does not satisfy the first inequality, Set 2 is invalid.
### Set 3: [tex]\((2,2), (1,-2), (0,2)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((1, -2)\)[/tex]:
- [tex]\[ y = -2 \geq -\frac{1}{3}(1) + 2 = \frac{5}{3} \rightarrow (\text{False}) \][/tex]
Since the point [tex]\((1, -2)\)[/tex] does not satisfy the first inequality, Set 3 is invalid.
### Set 4: [tex]\((2,2), (1,2), (2,0)\)[/tex]
1. For the point [tex]\((2, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(2) + 2 = \frac{4}{3} \rightarrow (\text{True}) \][/tex]
- [tex]\[ y = 2 < 2(2) + 3 = 7 \rightarrow (\text{True}) \][/tex]
2. For the point [tex]\((1, 2)\)[/tex]:
- [tex]\[ y = 2 \geq -\frac{1}{3}(1) + 2 = \frac{7}{3} \rightarrow (\text{False}) \][/tex]
Since the point [tex]\((1, 2)\)[/tex] does not satisfy the first inequality, Set 4 is invalid.
### Conclusion
Only Set 1: [tex]\((2,2), (3,1), (4,2)\)[/tex] satisfies both inequalities. Thus, the valid set of points is Set 1.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.
