Problem 38 - College Algebra

Question

Cities A and 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 condition is expressed as \(|x| \geq 100\).

Step 1: Understand the Problem

We need to express the condition that the distance \( x \) between two cities is at least 100 miles using absolute value notation. City A is at the origin, and city B is located at a point \( x \) on the east-west line.

Step 2: Recall Absolute Value Meaning

Recall that the absolute value \( |x| \) represents the distance from the number \( x \) to 0 on the number line. It ensures the expression is non-negative, as distance cannot be negative.

Step 3: Formulate the Inequality in Words

The problem states that the distance \( x \) is at least 100 miles. This translates to the mathematical inequality \( x \geq 100 \). However, city B could be either east or west of city A, so we need to account for \( x \) being positive or negative.

Step 4: Convert the Inequality to Absolute Value Notation

To express the inequality using absolute value, ensure that both positions are considered. Use \( |x| \geq 100 \) to indicate that the distance from city B to city A is at least 100 miles, regardless of its direction on the east-west line.

Key Concepts

Understanding Distance and Absolute Value Notation

In mathematics, distance refers to the amount of space between two points. It is always a non-negative number and can be measured in units such as meters, kilometers, or miles.
When dealing with mathematical exercises like this one, involving cities on a coordinate system, it’s crucial to apply the absolute value notation for distance.

• The absolute value, denoted as \(|x|\), represents the distance of a number \(x\) from zero on the number line.
• This notation is especially useful because it ensures that our distance calculations are always non-negative.
• For example, when the exercise states that the distance from city A to city B is at least 100 miles, we use the absolute value notation \(|x| \geq 100\).

This notation accommodates any direction along the east-west line, signifying the city B could be either east or west of city A while still ensuring the distance condition is met.

Explaining Inequalities in Absolute Value

Inequalities are used to express the range or limit of values that a variable can take. For example, if we state \(x \geq 100\), it means \(x\) is not less than 100. However, when expressing this using absolute value, we can cover scenarios both to the left and right of an origin point.

• To illustrate, \(|x| \geq 100\) implies that the distance \(x\) is at least 100 miles in either direction from the origin, meaning it doesn't matter if city B is to the west (negative direction) or to the east (positive direction) of city A.
• Thanks to absolute value notation, the inequality is able to accommodate these two potential positions without having to split the condition into two separate inequalities.
• This simplifies equations and provides a clearer picture when dealing with linear situations such as this one.

In practical terms, this helps in scenarios where only the magnitude of a quantity is relevant, not the direction, perfect for measuring distances on a number line.

Insight into Coordinate Systems and Their Application

A coordinate system is a method for specifying each point uniquely in a plane by a pair of numerical coordinates. These coordinates represent the position of a point on an axis, such as north-south or east-west, which is particularly useful in geography and navigation.

• In the context of the exercise, we referenced a simplified one-dimensional coordinate system where city A is situated at the origin (point 0), and city B is placed along the same line.
• This approach enables easy computation of distance between two points using numeric relationships and absolute value for expressing physical distance without considering geographical direction.
• The use of coordinate systems in conjunction with absolute value notation thus simplifies the complexities involved in problems related to positioning and movement along a line.

By understanding how these systems work, students can better grasp real-world mathematical applications, such as planning routes or determining potential travel distances.