Problem 23 - Precalculus: Functions and Graphs

Question

The two given numbers are coordinates of points \(A\) and \(B\), respectively, on a coordinate line. Express the indicated statement as an inequality involving the absolute value symbol. $$4, \quad x;$$ \(d(A, B)\) is not greater than 3

Step-by-Step Solution

Answer

The inequality is \(|x - 4| \leq 3\).

Step 1: Understand Coordinates of Points

The points given are 4 and \( x \) on a coordinate line. These represent points \( A \) and \( B \) respectively.

Step 2: Define the Distance Formula

The distance between point \( A \) at 4 and point \( B \) at \( x \) is represented as \( d(A, B) = |x - 4| \).

Step 3: Translate Verbal Statement into Inequality

The inequality "\( d(A, B) \) is not greater than 3" means that \( |x - 4| \leq 3 \).

Step 4: Verify the Inequality

Check if the inequality \( |x - 4| \leq 3 \) correctly represents "the distance is not greater than 3". The absolute value indicates distance, and the inequality sign 'less than or equal to' expresses 'not greater than.'

Key Concepts

Coordinate Geometry

Coordinate geometry is an essential branch of mathematics that involves studying geometric figures using a coordinate system. In this context, we are using a one-dimensional coordinate system known as a coordinate line or number line. This approach allows us to pinpoint the position of points using numbers.

When we say point A has coordinate 4, it means on this line, point A is located at 4 units from the origin. Meanwhile, if point B is at coordinate \( x \), it simply implies that point B's position is \( x \) units from the origin. This method of representing points is foundational for further calculations, as it turns abstract geometric ideas into something tangible using just numbers.

Distance Formula

The distance formula is a crucial concept in coordinate geometry, used to calculate the distance between two points on a plane or line. In the one-dimensional coordinate system, the distance between two points is straightforward. It is given by the absolute value of the difference between their coordinates.

For instance, when finding the distance between points A and B with coordinates 4 and \( x \), respectively, the distance is expressed mathematically as

• \( d(A, B) = |x - 4| \)

The absolute value notation is significant here. It ensures the result is always non-negative, as distance cannot be negative by definition. So, whether \( x \) is greater or less than 4, \( |x - 4| \) will give us a positive representation of how far apart the two points are on the number line.

Translation of Verbal Statements into Inequalities

Translating verbal statements into mathematical expressions is a vital skill, especially when dealing with inequalities. In our example, we had the statement: "the distance \( d(A, B) \) is not greater than 3".

Understanding this, the phrase "not greater than" implies "less than or equal to" (\( \leq \)). When expressed using the distance formula, it turns the statement into a mathematical inequality:

• \( |x - 4| \leq 3 \)

Here, \( |x - 4| \) represents the distance between points A and B, and \( \leq 3 \) translates literary conditions into a mathematical boundary. This inequality is now a tool we can use for further analysis or to determine allowable values of \( x \) that satisfy the condition outlined in the original statement. Translating such statements accurately helps us better understand and solve real-world problems.