To find which of the given ordered pairs satisfies both inequalities, we need to apply each pair to both inequalities and check if they hold true.
The inequalities are:
1. [tex]\( y > -2x + 3 \)[/tex]
2. [tex]\( y \leq x - 2 \)[/tex]
Let's test each of the given pairs:
1. Pair [tex]\((0, 0)\)[/tex]:
- For the first inequality: [tex]\( 0 > -2(0) + 3 \)[/tex]
- Simplifies to: [tex]\( 0 > 3 \)[/tex] (This is false)
- Since the first inequality is not satisfied, we don't need to check the second one.
2. Pair [tex]\((0, -1)\)[/tex]:
- First inequality: [tex]\( -1 > -2(0) + 3 \)[/tex]
- Simplifies to: [tex]\( -1 > 3 \)[/tex] (This is false)
- Since the first inequality is not satisfied, we don't need to check the second one.
3. Pair [tex]\((1, 1)\)[/tex]:
- First inequality: [tex]\( 1 > -2(1) + 3 \)[/tex]
- Simplifies to: [tex]\( 1 > 1 \)[/tex] (This is false)
- Since the first inequality is not satisfied, we don't need to check the second one.
4. Pair [tex]\((30, 0)\)[/tex]:
- First inequality: [tex]\( 0 > -2(30) + 3 \)[/tex]
- Simplifies to: [tex]\( 0 > -60 + 3 \)[/tex]
- Further simplifies to: [tex]\( 0 > -57 \)[/tex] (This is true)
- Second inequality: [tex]\( 0 \leq 30 - 2 \)[/tex]
- Simplifies to: [tex]\( 0 \leq 28 \)[/tex] (This is true)
- Since both inequalities are satisfied, this is a valid pair.
Based on this analysis, the pair [tex]\((30, 0)\)[/tex] satisfies both inequalities. Therefore, the ordered pair that makes both inequalities true is [tex]\((30, 0)\)[/tex].
