Understanding Absolute Value in Equations and Graphs

The absolute value of a number is a mathematical concept commonly used on a number line or graph, representing the distance from zero (modulus). To explain this visually, the absolute value can be compared to physical distance.

Think of your house as the starting point, zero, and steps to the west as negative numbers, while steps to the east represent positive numbers. Whether you travel a mile east or west, you're still the same distance from home.

Traveling west doesn’t imply a negative number of steps, and moving east doesn’t mean a positive number of steps either. Both directions reflect the same distance from the origin. Therefore, a mile in either direction holds the same absolute value.

What Does an Absolute Value Symbol Look Like?

Absolute value symbols consist of vertical bars placed on either side of a number. For example, instead of writing "the absolute value of five," you can simply represent it as |5|.

4 Key Properties of Absolute Values

Now that you've grasped the basic concept of absolute value from the distance example above, let's explore how the four key properties can help you quickly identify and define this mathematical concept in various situations.

1. Non-negativity

An absolute value is always either positive or zero, never negative.

2. Positive-definiteness

In the context of absolute value, positive definiteness means that the absolute value of a number is zero only when the number is explicitly written as zero |0|.

3. Multiplicativity

The absolute value of the product of two numbers equals the product of their absolute values due to how the negative sign is handled in absolute value. Let's examine this with the following example.

The absolute value of the multiplication of -3 and 5 equals 15:

The result of multiplying |-3| and |5| also gives 15:

4. Subadditivity

Subadditivity is often explained using the following expression:

This implies that the absolute value of the sum of two real numbers is less than or equal to the sum of their individual absolute values. Let's substitute some values to see how this works in practice:

Whether a is -2 or 2, |a| will always be 2. Similarly, no matter if b is -5 or 5, |b| will always be 5. Hence:

Now, let's examine the other side of the inequality: |a + b|. The expression a + b (before applying the absolute value) can unfold in several different ways:

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

The same applies to the second and third examples:

Let's revisit the original inequality: |a + b| ≤ |a| + |b|. Regardless of the values chosen for a and b, the absolute value of their sum (|a + b|) will always be less than or equal to the sum of their individual absolute values (|a| + |b|).

Solving Absolute Value Equations

Begin by isolating the absolute value expression from the rest of the equation. Determine if the value is positive or negative, and understand how this affects the numbers involved. Solve for the unknown and verify your solution by graphing it.

Absolute Value Function Graphs

In mathematics, it is crucial to provide proof for your work to ensure the accuracy of your answer. The graph of the standard absolute value function is represented by f(x) = |x|.

This equation can also be written as f(x) = x, if x ≥ 0 and -x, if x < 0 (as shown in the diagram at the start of this article).