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In mathematics, absolute value is a way to describe how far a number is from zero on the number line, regardless of direction. It helps in measuring distances between numbers without considering their sign.
We denote the absolute value of a number by placing it within vertical bars. For example, the absolute value of -5 is written as \(|-5|\), which equals 5.
To calculate the absolute value:

• If the number is positive or zero, the absolute value is the number itself.
• If the number is negative, the absolute value is the number without the negative sign.

For example: \(|-8| = 8\) and \(|3| = 3\).
When tackling problems involving distance on the number line, like the one given, absolute value helps us find how far apart two numbers are.

Comparing distances on the number line is crucial for understanding how close different numbers are to each other. When comparing two numbers, you can use absolute value to calculate the distances from a reference point.
For instance, in the exercise, we need to find which number, -2 or -7, is closer to -4. Distance is always a positive measure, which is why we use absolute values.

• First, find the distance from -2 to -4: \(|-2 - (-4)| = |-2 + 4| = |2| = 2\).
• Next, find the distance from -7 to -4: \(|-7 - (-4)| = |-7 + 4| = |3| = 3\).

Compare the distances:
\(|2| = 2\) and \(|3| = 3\).
Since 2 is less than 3, -2 is closer to -4 than -7.

Integers are a set of numbers that include all whole numbers and their negatives. They are represented without fractions or decimals.
Some examples of integers are: -9, -3, 0, 1, 7. Integers are positioned along the number line at equal intervals.
When solving problems involving integers:

• Always include their sign (negative or positive).
• Understand their position and relation to other integers on the number line.

In our exercise, integers -2 and -7 are compared based on their distance to another integer, -4. Understanding both negative and positive values and their distances is vital in these comparisons.
Remember, negative integers lie to the left of zero on the number line, and positive integers lie to the right of zero. This helps visualize and compare their relative positions and distances effectively.