To determine which ordered pair makes both inequalities true, let's analyze each pair step-by-step.
Given inequalities:
[tex]\[ y > -2x + 3 \][/tex]
[tex]\[ y \leq x - 2 \][/tex]
We need to check these conditions for the following ordered pairs: \((0,0)\), \((0,-1)\), \((1,1)\), and \((3,0)\).
1. For the ordered pair \((0,0)\):
- Check the first inequality: \(y > -2x + 3\)
- Substitute \(x = 0\) and \(y = 0\):
[tex]\[0 > -2(0) + 3 \Rightarrow 0 > 3\][/tex]
- This is false.
- Since the first inequality is not satisfied, we do not need to check the second one.
2. For the ordered pair \((0,-1)\):
- Check the first inequality: \(y > -2x + 3\)
- Substitute \(x = 0\) and \(y = -1\):
[tex]\[-1 > -2(0) + 3 \Rightarrow -1 > 3\][/tex]
- This is false.
- Since the first inequality is not satisfied, we do not need to check the second one.
3. For the ordered pair \((1,1)\):
- Check the first inequality: \(y > -2x + 3\)
- Substitute \(x = 1\) and \(y = 1\):
[tex]\[1 > -2(1) + 3 \Rightarrow 1 > 1\][/tex]
- This is false.
- Since the first inequality is not satisfied, we do not need to check the second one.
4. For the ordered pair \((3,0)\):
- Check the first inequality: \(y > -2x + 3\)
- Substitute \(x = 3\) and \(y = 0\):
[tex]\[0 > -2(3) + 3 \Rightarrow 0 > -6 + 3 \Rightarrow 0 > -3\][/tex]
- This is true.
- Check the second inequality: \(y \leq x - 2\)
- Substitute \(x = 3\) and \(y = 0\):
[tex]\[0 \leq 3 - 2 \Rightarrow 0 \leq 1\][/tex]
- This is true.
Since the ordered pair \((3,0)\) satisfies both inequalities, the answer is:
[tex]\[
\boxed{(3,0)}
\][/tex]
