- Identify the first condition: "less than 37" translates to \(r < 37\).
- Identify the second condition: "no less than 42" translates to \(r \geq 42\).
- Combine the two scenarios with the logical operator "or".
- The final inequality representing the route time is \(\boxed{r < 37 \text{ or } r \geq 42}\).
Explanation
- Analyzing the Scenarios
The problem describes the duration of a bus route, denoted as \(r\), based on the time the bus driver leaves. There are two distinct scenarios given:
- If the bus driver leaves by 7:00 a.m., the route duration is less than 37 minutes. This is represented by the inequality \(r < 37\).
- If the bus driver leaves after 7:00 a.m., the route duration is no less than 42 minutes. The phrase "no less than" means the duration must be greater than or equal to 42 minutes. This is represented by the inequality \(r \geq 42\).
Since these two scenarios are mutually exclusive depending on the time of departure, we connect them using the word "or," which corresponds to the logical union of the two sets of possible durations.
- Formulating the Final Inequality
We now combine the individual inequalities derived from the problem description:
- The first condition is \(r < 37\).
- The second condition is \(r \geq 42\).
Combining these using the logical "or" gives the final expression for the route time:
\[r < 37 \text{ or } r \geq 42\]
Comparing this result with the given choices, we see that it matches the first option.
Examples
Understanding how to translate word problems into mathematical inequalities is essential for data analysis and logistical planning. For instance, a delivery company might estimate that a delivery will take less than 20 minutes if the truck takes a main highway, but if traffic is heavy, it will take at least 50 minutes. Representing this as \(t < 20 \text{ or } t \geq 50\) helps planners allocate time resources and communicate expectations to customers effectively, ensuring that both typical and worst-case scenarios are accounted for in their scheduling systems.