Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 73-76: Express the statement in terms of an inequality involving an absolute value. The arrival time \(t\) of train B must be at least 5 minutes different from the 4:00 P.M. arrival time of train A.

Step-by-Step Solution

Answer

The inequality is \(|t - 240| \geq 5\).

Step 1: Identify Key Information

The information given is that the arrival time of train B, denoted as \( t \), must be at least 5 minutes different from 4:00 P.M., the arrival time of train A.

Step 2: Understand Minimum Difference Requirement

The phrase 'at least 5 minutes different' means that the absolute difference between the time \( t \) and 4:00 P.M. should be greater than or equal to 5 minutes.

Step 3: Define a Reference Point

Consider 4:00 P.M. as 240 minutes from midnight. Thus, the reference point for train A's arrival time in minutes is 240.

Step 4: Construct the Absolute Value Inequality

Since the absolute difference between train B's time \( t \) and the reference time 240 must be at least 5, we set up the inequality: \(|t - 240| \geq 5\).

Step 5: Explain in Words

This inequality states that train B's arrival time must be either 5 or more minutes before 4:00 P.M. or 5 or more minutes after 4:00 P.M.

Key Concepts

Train Arrival Time

Understanding the concept of train arrival time is crucial, especially when dealing with absolute value inequalities. Train arrival time refers to the specific time at which a train is scheduled or actually arrives at its designated station. This time can be expressed in various formats, such as in hours and minutes of a typical day.
In the context of our problem, we are dealing with the arrival time of train B. It's important because we need to compare it with a specific reference point or another train's arrival time, which is train A in this case. For such problems, it is often handy to convert the given time into a full number such as minutes from midnight, which allows for easier calculation when setting up inequalities.
Keep in mind that train schedules are typically precise, and using an accurate time reference allows us to solve problems involving time differences and absolute values efficiently.

Time Difference

The time difference between two events, like the arrival of two trains, is a critical concept. It refers to the duration or interval separating two points in time. When we talk about time differences in math, especially with inequalities, we often express it using absolute values to focus on magnitude rather than direction.
For our particular problem, the arrival time of train B must differ from that of train A by at least 5 minutes. In mathematical terms, this requirement can be captured by the inequality \(|t - 240| \geq 5\), where 240 is the time of train A in total minutes from midnight.
This inequality emphasizes that train B can arrive either before or after train A as long as the minimum time difference of 5 minutes is maintained. Expressing the time difference with absolute values is an effective way to cater to both possible scenarios of earlier or later arrivals.

Reference Point

A reference point is an essential element when working with absolute value inequalities involving times. It represents a fixed time that provides a baseline for comparisons. In our scenario, 4:00 P.M. is the reference point and is converted into 240 minutes past midnight to facilitate easier calculations.
Having the reference point allows us to set the ground for establishing the minimum required differences between the two train arrivals. It simplifies the problem by providing a specific number to work with rather than considering the hour and minute hands separately.
Understanding how to choose and utilize a reference point is important not only for solving problems related to time but also in various mathematical contexts. It serves as the anchor around which other values, like the arrival time of train B, orbit. Hence, the comparison we aim to achieve via the inequality \( |t - 240| \geq 5\) becomes manageable and straightforward.