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: -2 and 5.
2. Recall that the distance between two numbers \(a\) and \(b\) on the number line can be expressed as the absolute value of their difference: \(\left| a - b \right|\) or \(\left| b - a \right|\).
3. Write the distance between -2 and 5 as an absolute value expression: \(\left| -2 - 5 \right|\) or \(\left| 5 - (-2) \right|\).
4. Simplify inside the absolute value: \(\left| -2 - 5 \right| = \left| -7 \right|\) or \(\left| 5 + 2 \right| = \left| 7 \right|\).
5. Evaluate the absolute value: \(\left| -7 \right| = 7\) or \(\left| 7 \right| = 7\), which gives the distance between the two numbers.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative and is denoted by vertical bars, for example, |x|. This concept helps measure how far apart two numbers are without considering which is larger.

Distance Between Two Numbers

The distance between two numbers on the number line is the absolute value of their difference. For numbers a and b, the distance is |a - b|, which ensures the result is non-negative and reflects the actual gap between the points.

Evaluating Absolute Value Expressions

To find the distance, first write the absolute value expression representing the difference between the numbers. Then calculate the difference inside the bars and take its absolute value, converting any negative result to positive to get the final distance.