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A number line is a visual representation of numbers laid out in a straight line. Each point on the line corresponds to a real number, and it helps us understand the arrangement of numbers in terms of size and value.

Number lines are essential in understanding absolute values and distances between numbers. They provide a way to visualize how far numbers are from each other. For instance, on a number line, -6 would be to the left of 0, and 8 would be to the right of 0. This spatial feature allows us to measure the "distance" between two points, such as -6 and 8, by counting the "steps" or spaces between them.

In our example, you can easily visualize that to get from -6 to 8, you need to move a certain number of steps or units to the right. Thus, the number line isn't just a static representation but an active tool for seeing the relationship between different numbers.

When we talk about the distance between two numbers, we're really referring to how far apart those numbers are on the number line. This concept is useful in many areas of math and real life.

To find this distance, we use the absolute value of their difference. The absolute value always gives us a non-negative result, which makes sense because distance can't be negative.

For our numbers, -6 and 8, you calculate the distance by taking the absolute value of the difference:

• First, write the expression:
  \(|8 - (-6)|\) becomes \(|8 + 6|\) when simplified.
• Then evaluate the expression:
  \(|8 + 6| = |14| = 14\).

Therefore, the distance between -6 and 8 is represented by the absolute value of 14, which is simply 14.

Evaluating mathematical expressions involves performing operations to find a specific value. In this context, we're focusing on evaluating absolute value expressions to understand distances.

Absolute value expressions can be identified by the symbols \(|...|\). These symbols tell us to consider only the magnitude of the number, disregarding whether it is positive or negative.

• To evaluate \(|8 - (-6)|\), we start by simplifying the expression inside to \(|8 + 6|\).
• This simplified form, \(|14|\), tells us that we only care about the distance from zero, which is 14.
• Since distances on a number line are always positive, the absolute value of 14 stays 14.

As a result, evaluating expressions with absolute values is straightforward and always ensures you end up with a positive distance for the numbers you're interested in.