Answer :
- Calculate the remaining water after Will fills his bottle: $144 - 16 = 128$ ounces.
- Set up the inequality representing the water distribution: $16x \le 128$.
- Solve for $x$: $x \le 8$.
- Combine with the condition that $x > 0$ to get the final range: $0 < x \le 8$, so the answer is $\boxed{(0, 8]}$.
### Explanation
1. Initial Analysis
Let's analyze the problem. Will starts with a 144-ounce cooler of water. He uses 16 ounces for his own water bottle, leaving $144 - 16 = 128$ ounces. He then distributes the remaining water into 16 cups, with each cup receiving $x$ ounces. We need to find the possible values of $x$.
2. Setting up the Inequality
The total amount of water distributed into the 16 cups is $16x$ ounces. This amount must be less than or equal to the remaining water in the cooler, which is 128 ounces. This gives us the inequality $16x \le 128$.
3. Solving for x
To solve for $x$, we divide both sides of the inequality by 16:$$\frac{16x}{16} \le \frac{128}{16}$$$$x \le 8$$
4. Determining the Range of x
Since Will is putting water into the cups, the amount of water in each cup must be greater than 0. So, $x > 0$. Combining this with the previous inequality, we get $0 < x \le 8$. This means that each cup can contain any amount of water greater than 0 ounces and up to 8 ounces, inclusive.
5. Final Answer and Graphing
Therefore, the number of ounces of water, $x$, that Will could have put in each cup is greater than 0 and less than or equal to 8. In interval notation, this is $(0, 8]$. To graph this on a number line, we would draw a line segment from 0 to 8. There would be an open circle at 0 (since $x$ cannot be equal to 0) and a closed circle at 8 (since $x$ can be equal to 8).
### Examples
Understanding inequalities like this can help in various real-life scenarios. For instance, if you're distributing snacks among a group of friends, and you have a limited amount of each snack, you can use inequalities to determine the possible amounts each person can receive while ensuring everyone gets something and you don't exceed your supply. This applies to resource allocation, budgeting, and even planning events where you need to manage quantities effectively.
- Set up the inequality representing the water distribution: $16x \le 128$.
- Solve for $x$: $x \le 8$.
- Combine with the condition that $x > 0$ to get the final range: $0 < x \le 8$, so the answer is $\boxed{(0, 8]}$.
### Explanation
1. Initial Analysis
Let's analyze the problem. Will starts with a 144-ounce cooler of water. He uses 16 ounces for his own water bottle, leaving $144 - 16 = 128$ ounces. He then distributes the remaining water into 16 cups, with each cup receiving $x$ ounces. We need to find the possible values of $x$.
2. Setting up the Inequality
The total amount of water distributed into the 16 cups is $16x$ ounces. This amount must be less than or equal to the remaining water in the cooler, which is 128 ounces. This gives us the inequality $16x \le 128$.
3. Solving for x
To solve for $x$, we divide both sides of the inequality by 16:$$\frac{16x}{16} \le \frac{128}{16}$$$$x \le 8$$
4. Determining the Range of x
Since Will is putting water into the cups, the amount of water in each cup must be greater than 0. So, $x > 0$. Combining this with the previous inequality, we get $0 < x \le 8$. This means that each cup can contain any amount of water greater than 0 ounces and up to 8 ounces, inclusive.
5. Final Answer and Graphing
Therefore, the number of ounces of water, $x$, that Will could have put in each cup is greater than 0 and less than or equal to 8. In interval notation, this is $(0, 8]$. To graph this on a number line, we would draw a line segment from 0 to 8. There would be an open circle at 0 (since $x$ cannot be equal to 0) and a closed circle at 8 (since $x$ can be equal to 8).
### Examples
Understanding inequalities like this can help in various real-life scenarios. For instance, if you're distributing snacks among a group of friends, and you have a limited amount of each snack, you can use inequalities to determine the possible amounts each person can receive while ensuring everyone gets something and you don't exceed your supply. This applies to resource allocation, budgeting, and even planning events where you need to manage quantities effectively.
