In Exercises 67–74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. −2 and 5

Verified step by step guidance

1. Identify the two numbers given in the problem: -2 and 5.
2. To find the distance between two numbers on a number line, use the formula for distance: \(|a - b|\), where \(a\) and \(b\) are the two numbers.
3. Substitute the given numbers into the formula: \(|-2 - 5|\).
4. Simplify the expression inside the absolute value: \(-2 - 5 = -7\).
5. Evaluate the absolute value: \(|-7|\), which represents the distance between the numbers.

Key Concepts

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 by two vertical bars, for example, |x|. For any real number x, |x| is equal to x if x is positive or zero, and -x if x is negative. This concept is crucial for understanding how to express distances between numbers.

Distance Between Numbers

The distance between two numbers on the number line can be calculated using the absolute value of their difference. Specifically, the distance d between two numbers a and b is given by the formula d = |a - b|. This formula allows us to quantify how far apart the two numbers are, which is essential for solving problems involving distances.

Evaluating Absolute Value Expressions

Evaluating an absolute value expression involves substituting the values into the expression and simplifying it to find the numerical distance. For example, to find the distance between -2 and 5, we calculate |−2 - 5|, which simplifies to |-7| = 7. This process is important for obtaining the final numerical answer after expressing the distance using absolute value.