Find the given distances between points P, Q, R, and S on a number line, with coordinates -4, -1, 8, and 12, respectively. See Example 3.d (P, Q)

Verified step by step guidance

1. Identify the coordinates of points P and Q on the number line. Here, P is at -4 and Q is at -1.
2. Recall the formula for finding the distance between two points on a number line: \( d(P, Q) = |x_2 - x_1| \), where \( x_1 \) and \( x_2 \) are the coordinates of the points.
3. Substitute the coordinates of P and Q into the formula: \( d(P, Q) = |-1 - (-4)| \).
4. Simplify the expression inside the absolute value: \( |-1 + 4| \).
5. Calculate the absolute value to find the distance between points P and Q.

Key Concepts

Distance on a Number Line

The distance between two points on a number line is calculated using the absolute difference of their coordinates. For points P and Q with coordinates a and b, the distance d is given by d = |a - b|. This concept is fundamental in understanding how to measure the separation between any two points in a one-dimensional space.

Absolute Value

Absolute value is a mathematical function that measures the distance of a number from zero on the number line, regardless of direction. It is denoted as |x|, where x is any real number. This concept is crucial when calculating distances, as it ensures that the result is always non-negative, reflecting the actual distance between points.

Coordinates of Points

Coordinates are numerical values that define the position of points on a number line or in a coordinate system. In this context, points P, Q, R, and S have specific coordinates (-4, -1, 8, and 12, respectively) that allow us to identify their locations. Understanding how to interpret and manipulate these coordinates is essential for solving distance-related problems.