Absolutely! Let's analyze the conditions and the points step-by-step to determine which points satisfy both inequalities:
Given inequalities:
1. \( y \leq -x + 1 \)
2. \( y > x \)
Points to be evaluated:
- \((-3, 5)\)
- \((-2, 2)\)
- \((-1, -3)\)
- \((0, -1)\)
Step-by-step evaluation of each point:
1. Point \((-3, 5)\):
- Check \( y \leq -x + 1 \):
[tex]\[ 5 \leq -(-3) + 1 \][/tex]
[tex]\[ 5 \leq 4 \][/tex]
This is false.
- Since the first condition is not satisfied, this point cannot be in the solution set.
2. Point \((-2, 2)\):
- Check \( y \leq -x + 1 \):
[tex]\[ 2 \leq -(-2) + 1 \][/tex]
[tex]\[ 2 \leq 3 \][/tex]
This is true.
- Check \( y > x \):
[tex]\[ 2 > -2 \][/tex]
This is also true.
- Since both conditions are satisfied, this point is valid.
3. Point \((-1, -3)\):
- Check \( y \leq -x + 1 \):
[tex]\[ -3 \leq -(-1) + 1 \][/tex]
[tex]\[ -3 \leq 2 \][/tex]
This is true.
- Check \( y > x \):
[tex]\[ -3 > -1 \][/tex]
This is false.
- Since the second condition is not satisfied, this point cannot be in the solution set.
4. Point \((0, -1)\):
- Check \( y \leq -x + 1 \):
[tex]\[ -1 \leq -(0) + 1 \][/tex]
[tex]\[ -1 \leq 1 \][/tex]
This is true.
- Check \( y > x \):
[tex]\[ -1 > 0 \][/tex]
This is false.
- Since the second condition is not satisfied, this point cannot be in the solution set.
Conclusion:
After evaluating all points against the given conditions, the only point that meets both inequalities is \((-2, 2)\).
Hence, the valid point is:
[tex]\[
[(-2, 2)]
\][/tex]
