To determine which pairs [tex]\((x, y)\)[/tex] are reasonable solutions for the situation where [tex]\(x\)[/tex] represents the number of hot dogs sold and [tex]\(y\)[/tex] represents the number of bottles of water sold, we need to consider the following criteria:
1. Both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] should be non-negative integers because you can't sell a negative or fractional number of items.
Let’s evaluate each given pair according to these criteria:
1. [tex]\((-1, 5)\)[/tex]:
- [tex]\(x = -1\)[/tex], which is less than 0.
- Since [tex]\(x\)[/tex] is a negative number, this pair is not reasonable.
2. [tex]\((0, 6)\)[/tex]:
- [tex]\(x = 0\)[/tex], which is non-negative.
- [tex]\(y = 6\)[/tex], which is non-negative.
- Both values are non-negative integers, so this pair is reasonable.
3. [tex]\((2, 1)\)[/tex]:
- [tex]\(x = 2\)[/tex], which is non-negative.
- [tex]\(y = 1\)[/tex], which is non-negative.
- Both values are non-negative integers, so this pair is reasonable.
4. [tex]\((1, 1.5)\)[/tex]:
- [tex]\(x = 1\)[/tex], which is non-negative.
- [tex]\(y = 1.5\)[/tex], which is not an integer.
- Since [tex]\(y\)[/tex] is not an integer, this pair is not reasonable.
5. [tex]\((1, 3)\)[/tex]:
- [tex]\(x = 1\)[/tex], which is non-negative.
- [tex]\(y = 3\)[/tex], which is non-negative.
- Both values are non-negative integers, so this pair is reasonable.
6. [tex]\((2, 2)\)[/tex]:
- [tex]\(x = 2\)[/tex], which is non-negative.
- [tex]\(y = 2\)[/tex], which is non-negative.
- Both values are non-negative integers, so this pair is reasonable.
Therefore, the reasonable pairs for the situation are:
- [tex]\((0, 6)\)[/tex]
- [tex]\((2, 1)\)[/tex]
- [tex]\((1, 3)\)[/tex]
- [tex]\((2, 2)\)[/tex]
