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 (Q, R)

Verified step by step guidance

1. Identify the coordinates of points Q and R on the number line. Q has a coordinate of -1 and R has a coordinate of 8.
2. Recall the formula for finding the distance between two points on a number line: \( d(Q, R) = |x_2 - x_1| \), where \( x_1 \) and \( x_2 \) are the coordinates of the points.
3. Substitute the coordinates of Q and R into the formula: \( d(Q, R) = |8 - (-1)| \).
4. Simplify the expression inside the absolute value: \( 8 - (-1) = 8 + 1 \).
5. Calculate the absolute value: \( |8 + 1| \).

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 A and B 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 along a linear scale.

Absolute Value

Absolute value is a mathematical function that measures the magnitude of a number without regard to its sign. For any real number x, the absolute value is denoted as |x| and is defined as x if x is positive or zero, and -x if x is negative. This concept is crucial when calculating distances, as it ensures that the result is always a non-negative value.

Coordinate System

A coordinate system provides a framework for identifying the position of points in space using numbers. In a one-dimensional number line, each point corresponds to a real number, which represents its position. Understanding how to interpret coordinates is essential for solving problems related to distances between points, as it allows for accurate calculations based on their respective values.