Certainly! Let's determine which points from the given list satisfy both inequalities.
The provided points are: [tex]\((0, 0)\)[/tex], [tex]\((2, 5)\)[/tex], [tex]\((4, -1)\)[/tex], and [tex]\((1, 1)\)[/tex].
We need to check each point against the system of inequalities:
1. [tex]\( y > 3x - 3 \)[/tex]
2. [tex]\( y < \frac{1}{2}x - 1 \)[/tex]
Checking [tex]\((0, 0)\)[/tex]:
For the first inequality:
[tex]\[ 0 > 3(0) - 3 \][/tex]
[tex]\[ 0 > -3 \][/tex] (True)
For the second inequality:
[tex]\[ 0 < \frac{1}{2}(0) - 1 \][/tex]
[tex]\[ 0 < -1 \][/tex] (False)
Since it does not satisfy the second inequality, [tex]\((0, 0)\)[/tex] is not in the solution area.
Checking [tex]\((2, 5)\)[/tex]:
For the first inequality:
[tex]\[ 5 > 3(2) - 3 \][/tex]
[tex]\[ 5 > 6 - 3 \][/tex]
[tex]\[ 5 > 3 \][/tex] (True)
For the second inequality:
[tex]\[ 5 < \frac{1}{2}(2) - 1 \][/tex]
[tex]\[ 5 < 1 - 1 \][/tex]
[tex]\[ 5 < 0 \][/tex] (False)
Since it does not satisfy the second inequality, [tex]\((2, 5)\)[/tex] is not in the solution area.
Checking [tex]\((4, -1)\)[/tex]:
For the first inequality:
[tex]\[ -1 > 3(4) - 3 \][/tex]
[tex]\[ -1 > 12 - 3 \][/tex]
[tex]\[ -1 > 9 \][/tex] (False)
Since it does not satisfy the first inequality, [tex]\((4, -1)\)[/tex] is not in the solution area.
Checking [tex]\((1, 1)\)[/tex]:
For the first inequality:
[tex]\[ 1 > 3(1) - 3 \][/tex]
[tex]\[ 1 > 3 - 3 \][/tex]
[tex]\[ 1 > 0 \][/tex] (True)
For the second inequality:
[tex]\[ 1 < \frac{1}{2}(1) - 1 \][/tex]
[tex]\[ 1 < \frac{1}{2} - 1 \][/tex]
[tex]\[ 1 < -0.5 \][/tex] (False)
Since it does not satisfy the second inequality, [tex]\((1, 1)\)[/tex] is not in the solution area.
After evaluating all points, none of them satisfy both inequalities. Therefore, the set of points in the solution area is:
[tex]\[
\boxed{[]}
\][/tex]
