Problem 38 - College Algebra with Corequisite Support

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 \(\mathrm{A}\) to city \(\mathrm{B}\) is at least 100 miles and \(x\) represents the distance from city B to city \(\mathrm{A}\), express this using absolute value notation.

Step-by-Step Solution

Answer

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

Step 1: Identify Key Information

City \(A\) is at the origin on an east-west line, so its coordinate is 0. The distance \(x\) from city \(B\) to city \(A\) is given to be at least 100 miles. This means the distance is equal to or greater than 100 miles.

Step 2: Formulate the Absolute Value Expression

The absolute value notation is used to describe the distance between two points on a number line, which is non-negative. Therefore, the condition \(x \geq 100\) can be expressed using absolute value notation as \(|x| \geq 100\).

Step 3: Explanation of the Absolute Value

|x| represents the distance from city B to city A without consideration of direction. Since distance cannot be negative, the absolute value operation ensures this condition.

Key Concepts

Distance Problems

Distance problems often involve determining how far apart two points are. In this exercise, we are dealing with the distance between two cities that lie on the same east-west line.
When discussing such distances, the direction is generally not significant, which is why we use absolute value notation.
The key is understanding how to express the given conditions: City A is at the origin, and City B must be at least 100 miles away from City A. Distance can never be negative, so when we are asked to say that the distance from B to A is at least 100 miles, it means

• The distance is 100 miles or more.
• The distance between them could be greater than 100 miles.

To express this mathematically, we use \[|x| \geq 100\], because absolute value measures the magnitude of distance without worrying about which city is east or west on the number line. This concept is fundamental when solving many distance-related problems.

Coordinate System

The coordinate system is like a map that helps us understand the position of objects in space. In this problem, we are dealing with a one-dimensional coordinate system: the east-west line.
City A is located at the origin, which is the reference point having the coordinate value of 0.
City B is some unknown distance, represented by \(x\), away from city A. In a coordinate system, these cities' positions can be thought of as points on a number line.

• The origin (0) is where City A is placed.
• City B is represented by \(x\), located at a distance from the origin.

The coordinate system helps us to represent these distance facts visually and interpret them numerically. It brings an element of precision to spatial analysis, making it easier to calculate and visualize distances between points.

Expressions with Inequalities

Inequalities help us describe how one value compares to another. In this problem, the expression involving inequality is \(|x| \geq 100\).
It is used to say that the distance \(x\) must be at least 100 miles. When dealing with inequalities, particularly in distance problems, it’s important to recognize that:

• The inequality \(\geq\) indicates that the value can be equal to or greater than the compared quantity.
• Using absolute value ensures that the comparison is non-directional, only concerned with magnitude.

The expression \(|x| \geq 100\) implies that distance, irrespective of whether it is in the positive or negative direction on the number line, should meet or exceed the 100-mile benchmark. Inequalities allow for a range of values rather than a single fixed one, offering flexibility in descriptions and solutions. This principle is widely used in mathematical expressions and real-life scenarios alike, ensuring that conditions are met across a specified range.