How Absolute Value Works in Equations and Graphs

Now that you understand the general meaning of absolute value from the distance example above, read on to learn how you can use the following four fundamental properties to quickly identify and define a specific instance of this mathematical concept.

1. Non-negativity

An absolute value will always be a positive or non-negative value.

2. Positive-definiteness

In regards to absolute value, positive definiteness means that a number’s absolute value is zero only when the number is explicitly denoted as zero |0|.

3. Multiplicativity

The absolute value of the product of two numbers is the same as the product of the absolute values because of what the negative sign becomes in an absolute value. Let’s look at the following example.

The absolute value of the product of -3 and 5 is 15:

The product of |-3| and |5| is also 15:

4. Subadditivity

The concepts of subadditivity are typically illustrated in the following expression:

This means that the absolute value of the sum of two real numbers is less than or equal to the total of the absolute values of the two numbers. Let’s plug in some numbers to see how this plays out in a real example:

Whether a is -2 or 2, |a| is 2. Similarly, whether b is -5 or 5, |b| is 5. Therefore:

Now let’s look at the other side of that inequality: |a + b|. The expression a + b (before you take the absolute value of the expression) can play out any of the following ways:

The absolute value of the first and last examples is 7:

The same goes for the second and third examples:

Now, let’s go back to the initial inequality: |a + b| ≤ |a| + |b|. No matter what values you plug into a and b, you’ll find that the absolute value of the sum (|a + b|) is less than or equal to the sum of the absolute values (|a| + |b|).