Question

Lowell runs 1.2 miles daily. His goal is to run 120 miles, and he will end his daily runs when he reaches that goal. He uses the equation y = -1.2 + 120 to keep track of the distance he still needs to run over time. Complete the inequality to determine the domain of the function given the real world constraints of the problem

Answer 1

To determine the domain of the function based on the real-world constraints of Lowell's running, we need to analyze the given information:

1. **Total goal**: 120 miles
2. **Daily run**: 1.2 miles
3. **Equation provided**: \( y = -1.2x + 120 \), where \( y \) is the distance left to run and \( x \) is the number of days he has run.

### Constraints:
- Lowell can run for a maximum number of days until he reaches his goal of 120 miles.
- To find the maximum number of days (let's denote it as \( x_{\text{max}} \)), we set \( y \) (the distance left to run) to 0 because he stops running once he completes his goal:

\[
0 = -1.2x + 120
\]

Solving for \( x \):

\[
1.2x = 120
\]
\[
x = \frac{120}{1.2} = 100
\]

### Inequality for domain:
Given that \( x \) represents the number of days and must be non-negative, we have:

\[
0 \leq x \leq 100
\]

### Completed inequality:
Therefore, the domain of the function, considering the real-world constraints of the problem, is:

\[
0 \leq x \leq 100
\]

This means that \( x \) can take any value from 0 to 100, inclusive.