Answer :
To determine which ordered pairs satisfy both given inequalities, let's analyze each pair step-by-step.
### Inequalities:
1. [tex]\( x + y > 0 \)[/tex]
2. [tex]\( x - y < 5 \)[/tex]
Let's check each ordered pair against these inequalities.
#### Ordered Pair: [tex]\((-5, 5)\)[/tex]
1. [tex]\( x + y = -5 + 5 = 0 \)[/tex] (This does NOT satisfy [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = -5 - 5 = -10 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair does not satisfy the first inequality.
#### Ordered Pair: [tex]\((0, 3)\)[/tex]
1. [tex]\( x + y = 0 + 3 = 3 \)[/tex] (This satisfies [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 0 - 3 = -3 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair satisfies both inequalities.
#### Ordered Pair: [tex]\((0, -2)\)[/tex]
1. [tex]\( x + y = 0 - 2 = -2 \)[/tex] (This does NOT satisfy [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 0 - (-2) = 2 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair does not satisfy the first inequality.
#### Ordered Pair: [tex]\((1, 1)\)[/tex]
1. [tex]\( x + y = 1 + 1 = 2 \)[/tex] (This satisfies [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 1 - 1 = 0 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair satisfies both inequalities.
#### Ordered Pair: [tex]\((3, -4)\)[/tex]
1. [tex]\( x + y = 3 - 4 = -1 \)[/tex] (This does NOT satisfy [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 3 - (-4) = 7 \)[/tex] (This does NOT satisfy [tex]\( x - y < 5 \)[/tex])
This pair does not satisfy either inequality.
### Conclusion
From the pairs checked, the ordered pairs that make both inequalities true are:
- [tex]\((0, 3)\)[/tex]
- [tex]\((1, 1)\)[/tex]
Thus, the ordered pairs [tex]\((0, 3)\)[/tex] and [tex]\((1, 1)\)[/tex] are the ones that satisfy both given inequalities.
### Inequalities:
1. [tex]\( x + y > 0 \)[/tex]
2. [tex]\( x - y < 5 \)[/tex]
Let's check each ordered pair against these inequalities.
#### Ordered Pair: [tex]\((-5, 5)\)[/tex]
1. [tex]\( x + y = -5 + 5 = 0 \)[/tex] (This does NOT satisfy [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = -5 - 5 = -10 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair does not satisfy the first inequality.
#### Ordered Pair: [tex]\((0, 3)\)[/tex]
1. [tex]\( x + y = 0 + 3 = 3 \)[/tex] (This satisfies [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 0 - 3 = -3 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair satisfies both inequalities.
#### Ordered Pair: [tex]\((0, -2)\)[/tex]
1. [tex]\( x + y = 0 - 2 = -2 \)[/tex] (This does NOT satisfy [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 0 - (-2) = 2 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair does not satisfy the first inequality.
#### Ordered Pair: [tex]\((1, 1)\)[/tex]
1. [tex]\( x + y = 1 + 1 = 2 \)[/tex] (This satisfies [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 1 - 1 = 0 \)[/tex] (This satisfies [tex]\( x - y < 5 \)[/tex])
This pair satisfies both inequalities.
#### Ordered Pair: [tex]\((3, -4)\)[/tex]
1. [tex]\( x + y = 3 - 4 = -1 \)[/tex] (This does NOT satisfy [tex]\( x + y > 0 \)[/tex])
2. [tex]\( x - y = 3 - (-4) = 7 \)[/tex] (This does NOT satisfy [tex]\( x - y < 5 \)[/tex])
This pair does not satisfy either inequality.
### Conclusion
From the pairs checked, the ordered pairs that make both inequalities true are:
- [tex]\((0, 3)\)[/tex]
- [tex]\((1, 1)\)[/tex]
Thus, the ordered pairs [tex]\((0, 3)\)[/tex] and [tex]\((1, 1)\)[/tex] are the ones that satisfy both given inequalities.
