Answer :
To determine which ordered pair is a solution to the given system of inequalities:
[tex]\[ \begin{array}{l} y < -x^2 + x \\ y > x^2 - 4 \end{array} \][/tex]
we need to check each pair [tex]\((x, y)\)[/tex] against the inequalities individually.
1. Evaluating the pair [tex]\((0, -1)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ -1 < -(0)^2 + 0 \implies -1 < 0 \quad \text{(True)} \][/tex]
- For [tex]\(y > x^2 - 4\)[/tex]:
[tex]\[ -1 > (0)^2 - 4 \implies -1 > -4 \quad \text{(True)} \][/tex]
Since [tex]\((0, -1)\)[/tex] satisfies both inequalities, it is a solution to the system.
2. Evaluating the pair [tex]\((1, 1)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ 1 < -(1)^2 + 1 \implies 1 < 0 \quad \text{(False)} \][/tex]
Since [tex]\((1, 1)\)[/tex] does not satisfy the first inequality, it is not a solution.
3. Evaluating the pair [tex]\((2, -3)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ -3 < -(2)^2 + 2 \implies -3 < -2 \quad \text{(True)} \][/tex]
- For [tex]\(y > x^2 - 4\)[/tex]:
[tex]\[ -3 > (2)^2 - 4 \implies -3 > 0 \quad \text{(False)} \][/tex]
Since [tex]\((2, -3)\)[/tex] does not satisfy the second inequality, it is not a solution.
4. Evaluating the pair [tex]\((3, -6)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ -6 < -(3)^2 + 3 \implies -6 < -6 \quad \text{(False)} \][/tex]
Since [tex]\((3, -6)\)[/tex] does not satisfy the first inequality, it is not a solution.
Based on this evaluation, the only ordered pair that satisfies both inequalities is [tex]\((0, -1)\)[/tex]. Therefore, the solution to the given system of inequalities is:
[tex]\[ (0, -1) \][/tex]
[tex]\[ \begin{array}{l} y < -x^2 + x \\ y > x^2 - 4 \end{array} \][/tex]
we need to check each pair [tex]\((x, y)\)[/tex] against the inequalities individually.
1. Evaluating the pair [tex]\((0, -1)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ -1 < -(0)^2 + 0 \implies -1 < 0 \quad \text{(True)} \][/tex]
- For [tex]\(y > x^2 - 4\)[/tex]:
[tex]\[ -1 > (0)^2 - 4 \implies -1 > -4 \quad \text{(True)} \][/tex]
Since [tex]\((0, -1)\)[/tex] satisfies both inequalities, it is a solution to the system.
2. Evaluating the pair [tex]\((1, 1)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ 1 < -(1)^2 + 1 \implies 1 < 0 \quad \text{(False)} \][/tex]
Since [tex]\((1, 1)\)[/tex] does not satisfy the first inequality, it is not a solution.
3. Evaluating the pair [tex]\((2, -3)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ -3 < -(2)^2 + 2 \implies -3 < -2 \quad \text{(True)} \][/tex]
- For [tex]\(y > x^2 - 4\)[/tex]:
[tex]\[ -3 > (2)^2 - 4 \implies -3 > 0 \quad \text{(False)} \][/tex]
Since [tex]\((2, -3)\)[/tex] does not satisfy the second inequality, it is not a solution.
4. Evaluating the pair [tex]\((3, -6)\)[/tex]:
- For [tex]\(y < -x^2 + x\)[/tex]:
[tex]\[ -6 < -(3)^2 + 3 \implies -6 < -6 \quad \text{(False)} \][/tex]
Since [tex]\((3, -6)\)[/tex] does not satisfy the first inequality, it is not a solution.
Based on this evaluation, the only ordered pair that satisfies both inequalities is [tex]\((0, -1)\)[/tex]. Therefore, the solution to the given system of inequalities is:
[tex]\[ (0, -1) \][/tex]
