Problem 38 - College Algebra

Question

Cities \(\mathrm{A}\) and \(\mathrm{B}\) are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and \(x\) represents the distance from city \(B\) to city \(A\) express this using absolute value notation.

Step-by-Step Solution

Answer

The expression is \(|x| \geq 100\).

Step 1: Understand the Problem

We need to find an equation that describes the condition that City B is at least 100 miles away from City A, given that City A is at the origin.

Step 2: Define the Distance from A to B

Since city A is located at the origin, the distance from city A to city B can be represented as \(|x|\), where \(x\) is the distance of city B from city A along the east-west line.

Step 3: Interpret 'At Least' in Mathematical Terms

The phrase 'at least 100 miles' translates to the inequality \(|x| \geq 100\). This inequality states that the distance \(x\) must be 100 miles or more.

Step 4: Final Step: Write the Expression Using Absolute Value

Putting it all together, the absolute value expression that describes the condition is \(|x| \geq 100\). This expression captures the requirement that city B is at least 100 miles away from city A.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions used to compare two values or expressions. Unlike equations, which show equality, inequalities indicate that one value is larger or smaller than another. In this problem, we encounter the inequality

• \(|x| \geq 100\)

This inequality shows that the absolute value of the distance \(x\) between cities A and B must be at least 100 miles. The \(\geq\) symbol represents 'greater than or equal to', setting a minimum distance condition.
This type of inequality shows flexibility, as it includes all distances equal to and greater than 100. Incorporating inequalities helps in setting boundaries in real-world problems, giving a range of possibilities rather than a fixed number.

Distance and Location on a Number Line

In this problem, distance plays a critical role. Distance is a non-negative value between two points. Here, cities A and B are points on an east-west line. City A is at the origin, meaning at point 0, while city B can be any point 100 units or more away from city A.
To express this distance, we use absolute value, notated as \(|x|\). This ensures that we measure distance without regard to direction, focusing only on how far apart they are, not in which direction. The absolute value term accounts for this by always yielding a non-negative result.
Understanding distance in terms of location on a number line helps in visualizing problems. It allows us to see how cities or points exist relative to each other, providing a clear mental picture of their separation.

The East-West Line Concept

The east-west line is a conceptual straight line often used to simplify distance problems. In this problem, cities A and B lie on this line, simplifying their position to one dimension. This approach typically assumes no north-south variation, making calculations easier.
By putting everything on a line, you reduce the complexity of the problem, focusing only on the positions along this particular axis. In this scenario, the east-west line implies all movement and distances are linear and only vary in this horizontal direction.

• The city at the origin (0) is a key starting point for measuring.
• Distances are measured east or west on this line.

This concept is a well-used method in problems where the direction of measurement is the primary interest, providing an essential way to break down geographic locations into simpler, one-dimensional problems.