# Absolute Values – An Introduction

This article will be about absolute values. This topic is somewhat tricky for those new to this topic.

The Absolute Value

The absolute value of a real number is the non-negative part of . It is expressed as . Sometimes it is referred to the modulus and can be thought of as a distance from zero.

The simplest case of absolute values is with numbers. Here are some examples.

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So far we have dealt with absolute value of numbers and not variables. We now generalize and present the formal definition.

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Here is a visual of the absolute function centered at .

Source: https://www.mathsisfun.com/sets/images/function-absolute.gif

The absolute function is an example of a piecewise function. This type of function depends on the value of . It is not like linear functions where it is a smooth line for all values. If we had then the picture above would be upside down and would look like a hat.

We now consider intercepts with the variable inside the absolute value. If we had then the origin point at (0,0) from would be shifted to the right 7 units to the point (7, 0) for . If we had then the origin point at (0,0) from would be shifted to the left 2 units to the point (2, 0) for . Refer to the picture below as a visual aid.

Any vertical shifts would have numbers outside the absolute value function. For example, would shift the every point from up by 3 units.

Domain and Range

For the absolute function , the domain is all real numbers for . The range for is for all positive numbers and zero.

Properties

There are properties associated with the absolute value function. Here are a few.

These two properties come in handy when dealing with inequalities. (The constant is a real number.)

For example if we have , we can solve for as follows:

Here is another example where we have . The second property is used.

Notes

There are more things that can be said about the absolute value of . The topic of absolute values and complex numbers are not mentioned here.