Let's analyze each point [tex]\((-3, 5)\)[/tex], [tex]\((-2, 2)\)[/tex], and [tex]\((-1, -3)\)[/tex] against the given inequalities:
1. First Inequality: [tex]\( y \leq -x + 1 \)[/tex]
2. Second Inequality: [tex]\( y > x \)[/tex]
### Point: [tex]\((-3, 5)\)[/tex]
- First Inequality: [tex]\( y \leq -x + 1 \)[/tex]
Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 5 \)[/tex]:
[tex]\[
5 \leq -(-3) + 1 \implies 5 \leq 3 + 1 \implies 5 \leq 4 \quad \text{(False)}
\][/tex]
Since this point does not satisfy the first inequality, we do not need to check the second one.
### Point: [tex]\((-2, 2)\)[/tex]
- First Inequality: [tex]\( y \leq -x + 1 \)[/tex]
Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 2 \)[/tex]:
[tex]\[
2 \leq -(-2) + 1 \implies 2 \leq 2 + 1 \implies 2 \leq 3 \quad \text{(True)}
\][/tex]
The point satisfies the first inequality.
- Second Inequality: [tex]\( y > x \)[/tex]
Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 2 \)[/tex]:
[tex]\[
2 > -2 \quad \text{(True)}
\][/tex]
The point also satisfies the second inequality.
Since this point satisfies both inequalities, we record [tex]\(1\)[/tex] for this point.
### Point: [tex]\((-1, -3)\)[/tex]
- First Inequality: [tex]\( y \leq -x + 1 \)[/tex]
Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[
-3 \leq -(-1) + 1 \implies -3 \leq 1 + 1 \implies -3 \leq 2 \quad \text{(True)}
\][/tex]
The point satisfies the first inequality.
- Second Inequality: [tex]\( y > x \)[/tex]
Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[
-3 > -1 \quad \text{(False)}
\][/tex]
Since this point does not satisfy the second inequality, we record [tex]\(0\)[/tex] for this point.
### Summary
- For the point [tex]\((-3, 5)\)[/tex], neither inequality is satisfied → 0
- For the point [tex]\((-2, 2)\)[/tex], both inequalities are satisfied → 1
- For the point [tex]\((-1, -3)\)[/tex], only one inequality is satisfied → 0
Therefore, the final result for the points [tex]\((-3, 5)\)[/tex], [tex]\((-2, 2)\)[/tex], and [tex]\((-1, -3)\)[/tex] is:
[tex]\[ [0, 1, 0] \][/tex]
