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 x is the non-negative part of x. It is expressed as |x|. 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.

\displaystyle |3| = 3

\displaystyle |-3| = 3

\displaystyle |-120| = 120

\displaystyle - |120| = - 120

\displaystyle |0| = 0

\displaystyle |-1 \cdot 0| = |0| = 0

\displaystyle |(-2)^2 | = |4| =4

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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.

\displaystyle |x| = \begin{cases}x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

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Here is a visual of the absolute function centered at x = 0.

 

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 x. It is not like linear functions where it is a smooth line for all x values. If we had -|x| then the picture above would be upside down and would look like a hat.

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

absolute

 

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


Domain and Range

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


Properties

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

\displaystyle |x| \geq 0

\displaystyle |x| = 0 \text{ if and only if (iff) } x = 0

\displaystyle |x \cdot y| = |x| \cdot |y|

\displaystyle |x + y| \leq |x| + |y| \hspace{0.5in} \text{ (Triangle Inequality)}

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

\displaystyle |x| \leq a \iff -a \leq x \leq a

\displaystyle |x| \geq a \iff x \geq a \text{ or } x \leq -a

For example if we have |x - 5| < 2, we can solve for x as follows:

\displaystyle |x - 5| < 2

\displaystyle -2 < x - 5 < 2

\displaystyle -2 + 5 < x < 2 + 5

\displaystyle 3 < x < 7

Here is another example where we have |x - 1| \geq 3. The second property is used.

\displaystyle |x - 1| \geq 3

\displaystyle (x - 1) \geq 3 \text{ or } (x - 1) \leq -3

\displaystyle x \geq 4 \text{ or } x \leq -2


Notes

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

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