To determine which ordered pair [tex]\((x, y)\)[/tex] satisfies both inequalities, we need to check each one step by step. The inequalities are:
1. [tex]\( y > -2x + 3 \)[/tex]
2. [tex]\( y \leq x - 2 \)[/tex]
Let's test each given pair:
1. Pair [tex]\((0,0)\)[/tex]:
- Inequality 1: [tex]\( 0 > -2(0) + 3 \Rightarrow 0 > 3\)[/tex] (False)
- Since it fails the first inequality, no need to check the second one.
2. Pair [tex]\((0,-1)\)[/tex]:
- Inequality 1: [tex]\( -1 > -2(0) + 3 \Rightarrow -1 > 3\)[/tex] (False)
- Since it fails the first inequality, no need to check the second one.
3. Pair [tex]\((1,1)\)[/tex]:
- Inequality 1: [tex]\( 1 > -2(1) + 3 \Rightarrow 1 > 1\)[/tex] (False)
- Since it fails the first inequality, no need to check the second one.
4. Pair [tex]\((3,0)\)[/tex]:
- Inequality 1: [tex]\( 0 > -2(3) + 3 \Rightarrow 0 > -6 + 3 \Rightarrow 0 > -3\)[/tex] (True)
- Since it satisfies the first inequality, we check the second:
- Inequality 2: [tex]\( 0 \leq 3 - 2 \Rightarrow 0 \leq 1\)[/tex] (True)
Therefore, the ordered pair [tex]\((3, 0)\)[/tex] makes both inequalities true. The solution is:
[tex]\[
(3, 0)
\][/tex]
