\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Absolute Value

Single Variable Calculus
Multi-Variable Calculus

Absolute values in functions are not hard to handle. You just need to get used to the notation and learn how to break up a function into different parts.

If you want a full lecture on absolute value equations and inequalities as well as other types of inequalities, here is a video clip.

freeCodeCamp.org - [39mins-18secs]

video by freeCodeCamp.org

What Does 'Absolute Value' Mean?

The absolute value signs are two vertical lines on both sides of an expression, for example \(\abs{x+3}\). When an expression is contained in a set of absolute value signs the idea is that the sign of the expression is stripped off and replaced with a positive sign. There are two possible cases.
[ Case 1 ] If the expression is already positive or zero, the absolute value signs do nothing and can be dropped.
[ Case 2 ] If the expression is less than zero, then the expression is replaced by the negative of itself and the absolute value signs can then be dropped.

Example 1: \(\abs{3} = 3\)
Since \(3\) is already positive, the absolute value signs do nothing and therefore can be dropped with no change in the expression, in this case \(3\).
Example 2: \(\abs{-3} = 3\)
In this case, the expression \(-3\) is stripped of the negative sign and replaced with the positive version of the expression, in this case the new value is \(3\).

These examples are pretty easy to see what needs to be done with the sign of the expression but most times, we do not know if the expression is positive or negative.
Example 3: \(\abs{x}\)
Since \(x\) is a variable and it's value has not been limited as positive, negative or even real, we can do nothing about the absolute value signs. The absolute value signs are a shorthand way of saying that when \(x\) is positive, the expression is just \(x\) but when \(x\) is negative, the expression is \(-x\).

This video clip explains this idea in more detail very well and shows a great way to write the absolute value function as a piecewise function.

Dr Chris Tisdell - What is the absolute value function? (part 1) [1min-57secs]

video by Dr Chris Tisdell

Notice he said in the video that the absolute value tells us the distance between things. This is a great way of looking at it.

Graphs of Absolute Value Functions

Let's watch a bit more of the same video above discussing how to graph absolute value functions and how to visualize the distance idea on the graph.

Dr Chris Tisdell - What is the absolute value function? (part 2) [about 9min]

video by Dr Chris Tisdell

For more information about the graphing of absolute value functions, see this page.

Okay, now let's learn how to solve absolute value equations on the next page.

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What Does 'Absolute Value' Mean?

Graphs of Absolute Value Functions


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