Answer :
To determine which ordered pair makes both inequalities true, we need to check each pair against the given inequalities:
[tex]\[ \begin{array}{l} (1) \: y > -2x + 3 \\ (2) \: y \leq x - 2 \end{array} \][/tex]
Let's evaluate each pair step-by-step:
1. Pair [tex]\((-1, 1)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ 1 > -2(-1) + 3 \][/tex]
[tex]\[ 1 > 2 + 3 \][/tex]
[tex]\[ 1 > 5 \quad \text{(False)} \][/tex]
- Since the first inequality is not satisfied, [tex]\((-1, 1)\)[/tex] does not satisfy both inequalities.
2. Pair [tex]\((0, -1)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ -1 > -2(0) + 3 \][/tex]
[tex]\[ -1 > 3 \quad \text{(False)} \][/tex]
- Since the first inequality is not satisfied, [tex]\((0, -1)\)[/tex] does not satisfy both inequalities.
3. Pair [tex]\((1, -1)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ -1 > -2(1) + 3 \][/tex]
[tex]\[ -1 > -2 + 3 \][/tex]
[tex]\[ -1 > 1 \quad \text{(False)} \][/tex]
- Since the first inequality is not satisfied, [tex]\((1, -1)\)[/tex] does not satisfy both inequalities.
4. Pair [tex]\((2, 0)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ 0 > -2(2) + 3 \][/tex]
[tex]\[ 0 > -4 + 3 \][/tex]
[tex]\[ 0 > -1 \quad \text{(True)} \][/tex]
- For the second inequality: [tex]\( y \leq x - 2 \)[/tex]
[tex]\[ 0 \leq 2 - 2 \][/tex]
[tex]\[ 0 \leq 0 \quad \text{(True)} \][/tex]
- Since both inequalities are satisfied, [tex]\((2, 0)\)[/tex] does satisfy both inequalities.
Therefore, the ordered pair [tex]\((2, 0)\)[/tex] makes both inequalities true.
[tex]\[ \begin{array}{l} (1) \: y > -2x + 3 \\ (2) \: y \leq x - 2 \end{array} \][/tex]
Let's evaluate each pair step-by-step:
1. Pair [tex]\((-1, 1)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ 1 > -2(-1) + 3 \][/tex]
[tex]\[ 1 > 2 + 3 \][/tex]
[tex]\[ 1 > 5 \quad \text{(False)} \][/tex]
- Since the first inequality is not satisfied, [tex]\((-1, 1)\)[/tex] does not satisfy both inequalities.
2. Pair [tex]\((0, -1)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ -1 > -2(0) + 3 \][/tex]
[tex]\[ -1 > 3 \quad \text{(False)} \][/tex]
- Since the first inequality is not satisfied, [tex]\((0, -1)\)[/tex] does not satisfy both inequalities.
3. Pair [tex]\((1, -1)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ -1 > -2(1) + 3 \][/tex]
[tex]\[ -1 > -2 + 3 \][/tex]
[tex]\[ -1 > 1 \quad \text{(False)} \][/tex]
- Since the first inequality is not satisfied, [tex]\((1, -1)\)[/tex] does not satisfy both inequalities.
4. Pair [tex]\((2, 0)\)[/tex]:
- For the first inequality: [tex]\( y > -2x + 3 \)[/tex]
[tex]\[ 0 > -2(2) + 3 \][/tex]
[tex]\[ 0 > -4 + 3 \][/tex]
[tex]\[ 0 > -1 \quad \text{(True)} \][/tex]
- For the second inequality: [tex]\( y \leq x - 2 \)[/tex]
[tex]\[ 0 \leq 2 - 2 \][/tex]
[tex]\[ 0 \leq 0 \quad \text{(True)} \][/tex]
- Since both inequalities are satisfied, [tex]\((2, 0)\)[/tex] does satisfy both inequalities.
Therefore, the ordered pair [tex]\((2, 0)\)[/tex] makes both inequalities true.
