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)

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Identify the coordinates of points P and Q on the number line. Here, P = -4 and Q = -1.

Recall that the distance between two points on a number line is the absolute value of the difference of their coordinates. The formula is: \(d(P, Q) = |x_Q - x_P|\).

Substitute the coordinates of P and Q into the formula: \(d(P, Q) = |-1 - (-4)|\).

Simplify the expression inside the absolute value: \(-1 - (-4) = -1 + 4\).

Calculate the absolute value to find the distance: \(d(P, Q) = |3|\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distance on a Number Line

The distance between two points on a number line is the absolute value of the difference between their coordinates. This ensures the distance is always a non-negative value, representing the length between points regardless of direction.

Absolute Value

Absolute value measures the magnitude of a real number without regard to its sign. It is used to find the distance between points by converting any negative difference into a positive number, which is essential for calculating distances on a number line.

Coordinate Representation of Points

Points on a number line are represented by their coordinates, which are real numbers indicating their position relative to zero. Understanding how to interpret these coordinates is crucial for calculating distances and solving related problems.

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