Most people think of the absolute value as using the positive value of a number. For example, |-5| = 5 and |17| = 17. So what’s so hard about absolute value? Many times, difficulty occurs when working with the formal definition or description.

Sometimes absolute value of a number is described as the distance from the origin to the number on a number line.

The formal definition says:

If a is a real number, the absolute value of a, denoted as |a|, is

The description of absolute value as a distance and the formal definition are both describing exactly what most people think of for absolute value.

Using distance as a way to describe absolute value just means that the answer will never be negative since distance is never going to be negative.

The formal definition is just a lengthy way of saying the following:

• If your number is positive or zero () then use that number for your answer.
• If your number is negative ( ) then use the opposite of that number for your answer.

Let's Practice

1. |-12| =

Since the number inside the absolute value symbol is negative, we change the sign and the answer is 12.

2. |0| =

Since the number inside the absolute value symbol is greater than or equal to zero, we use the number as it is so the answer is 0.

3. |3 - p| =

Since the value of is negative, we change the sign of the quantity inside the absolute value sign and get . You could get a decimal approximation for this number, but most textbooks and teachers would prefer this exact answer.