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Sagot :
Alright, let's determine which ordered pairs are in the solution set of the given system of linear inequalities.
The system of inequalities is:
1. [tex]\( y \geq -\frac{1}{2}x \)[/tex]
2. [tex]\( y < \frac{1}{2}x + 1 \)[/tex]
We will evaluate each ordered pair given in the four groups to see if it satisfies both inequalities. Let's take them one by one.
### Group 1:
[tex]\((5, -2), (3, 1), (-4, 2)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ -2 \geq -\frac{1}{2} \times 5 \][/tex]
[tex]\[ -2 \geq -2.5 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ -2 < \frac{1}{2} \times 5 + 1 \][/tex]
[tex]\[ -2 < 3.5 \][/tex] (True)
#### For [tex]\((3, 1)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 1 \geq -\frac{1}{2} \times 3 \][/tex]
[tex]\[ 1 \geq -1.5 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 1 < \frac{1}{2} \times 3 + 1 \][/tex]
[tex]\[ 1 < 2.5 \][/tex] (True)
#### For [tex]\((-4, 2)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 2 \geq -\frac{1}{2} \times -4 \][/tex]
[tex]\[ 2 \geq 2 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 2 < \frac{1}{2} \times -4 + 1 \][/tex]
[tex]\[ 2 < -1 + 1 \][/tex]
[tex]\[ 2 < 0 \][/tex] (False)
### Conclusion for Group 1:
The pair [tex]\((-4, 2)\)[/tex] does not satisfy both inequalities.
### Group 2:
[tex]\((5, -2), (3, -1), (4, -3)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- As before: (True, True)
#### For [tex]\((3, -1)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ -1 \geq -\frac{1}{2} \times 3 \][/tex]
[tex]\[ -1 \geq -1.5 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ -1 < \frac{1}{2} \times 3 + 1 \][/tex]
[tex]\[ -1 < 2.5 \][/tex] (True)
#### For [tex]\((4, -3)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ -3 \geq -\frac{1}{2} \times 4 \][/tex]
[tex]\[ -3 \geq -2 \][/tex] (False)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ -3 < \frac{1}{2} \times 4 + 1 \][/tex]
[tex]\[ -3 < 3 \][/tex] (True)
### Conclusion for Group 2:
The pair [tex]\((4, -3)\)[/tex] does not satisfy the first inequality.
### Group 3:
[tex]\((5, -2), (3, 1), (4, 2)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- As before: (True, True)
#### For [tex]\((3, 1)\)[/tex]:
- As before: (True, True)
#### For [tex]\((4, 2)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 2 \geq -\frac{1}{2} \times 4 \][/tex]
[tex]\[ 2 \geq -2 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 2 < \frac{1}{2} \times 4 + 1 \][/tex]
[tex]\[ 2 < 3 \][/tex] (True)
### Conclusion for Group 3:
All pairs satisfy both inequalities.
### Group 4:
[tex]\((5, -2), (-3, 1), (4, 2)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- As before: (True, True)
#### For [tex]\((-3, 1)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 1 \geq -\frac{1}{2} \times -3 \][/tex]
[tex]\[ 1 \geq 1.5 \][/tex] (False)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 1 < \frac{1}{2} \times -3 + 1 \][/tex]
[tex]\[ 1 < -1.5 + 1 \][/tex]
[tex]\[ 1 < -0.5 \][/tex] (False)
#### For [tex]\((4, 2)\)[/tex]:
- As before: (True, True)
### Conclusion for Group 4:
The pair [tex]\((-3, 1)\)[/tex] does not satisfy either inequality.
### Final Conclusion:
The group where all ordered pairs satisfy both inequalities is Group 3:
[tex]\((5, -2), (3, 1), (4, 2)\)[/tex].
The system of inequalities is:
1. [tex]\( y \geq -\frac{1}{2}x \)[/tex]
2. [tex]\( y < \frac{1}{2}x + 1 \)[/tex]
We will evaluate each ordered pair given in the four groups to see if it satisfies both inequalities. Let's take them one by one.
### Group 1:
[tex]\((5, -2), (3, 1), (-4, 2)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ -2 \geq -\frac{1}{2} \times 5 \][/tex]
[tex]\[ -2 \geq -2.5 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ -2 < \frac{1}{2} \times 5 + 1 \][/tex]
[tex]\[ -2 < 3.5 \][/tex] (True)
#### For [tex]\((3, 1)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 1 \geq -\frac{1}{2} \times 3 \][/tex]
[tex]\[ 1 \geq -1.5 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 1 < \frac{1}{2} \times 3 + 1 \][/tex]
[tex]\[ 1 < 2.5 \][/tex] (True)
#### For [tex]\((-4, 2)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 2 \geq -\frac{1}{2} \times -4 \][/tex]
[tex]\[ 2 \geq 2 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 2 < \frac{1}{2} \times -4 + 1 \][/tex]
[tex]\[ 2 < -1 + 1 \][/tex]
[tex]\[ 2 < 0 \][/tex] (False)
### Conclusion for Group 1:
The pair [tex]\((-4, 2)\)[/tex] does not satisfy both inequalities.
### Group 2:
[tex]\((5, -2), (3, -1), (4, -3)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- As before: (True, True)
#### For [tex]\((3, -1)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ -1 \geq -\frac{1}{2} \times 3 \][/tex]
[tex]\[ -1 \geq -1.5 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ -1 < \frac{1}{2} \times 3 + 1 \][/tex]
[tex]\[ -1 < 2.5 \][/tex] (True)
#### For [tex]\((4, -3)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ -3 \geq -\frac{1}{2} \times 4 \][/tex]
[tex]\[ -3 \geq -2 \][/tex] (False)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ -3 < \frac{1}{2} \times 4 + 1 \][/tex]
[tex]\[ -3 < 3 \][/tex] (True)
### Conclusion for Group 2:
The pair [tex]\((4, -3)\)[/tex] does not satisfy the first inequality.
### Group 3:
[tex]\((5, -2), (3, 1), (4, 2)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- As before: (True, True)
#### For [tex]\((3, 1)\)[/tex]:
- As before: (True, True)
#### For [tex]\((4, 2)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 2 \geq -\frac{1}{2} \times 4 \][/tex]
[tex]\[ 2 \geq -2 \][/tex] (True)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 2 < \frac{1}{2} \times 4 + 1 \][/tex]
[tex]\[ 2 < 3 \][/tex] (True)
### Conclusion for Group 3:
All pairs satisfy both inequalities.
### Group 4:
[tex]\((5, -2), (-3, 1), (4, 2)\)[/tex]
#### For [tex]\((5, -2)\)[/tex]:
- As before: (True, True)
#### For [tex]\((-3, 1)\)[/tex]:
- Check [tex]\( y \geq -\frac{1}{2}x \)[/tex]:
[tex]\[ 1 \geq -\frac{1}{2} \times -3 \][/tex]
[tex]\[ 1 \geq 1.5 \][/tex] (False)
- Check [tex]\( y < \frac{1}{2}x + 1 \)[/tex]:
[tex]\[ 1 < \frac{1}{2} \times -3 + 1 \][/tex]
[tex]\[ 1 < -1.5 + 1 \][/tex]
[tex]\[ 1 < -0.5 \][/tex] (False)
#### For [tex]\((4, 2)\)[/tex]:
- As before: (True, True)
### Conclusion for Group 4:
The pair [tex]\((-3, 1)\)[/tex] does not satisfy either inequality.
### Final Conclusion:
The group where all ordered pairs satisfy both inequalities is Group 3:
[tex]\((5, -2), (3, 1), (4, 2)\)[/tex].
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