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The algebraic definition of absolute value uses a piecewise-defined function.
A piecewise-defined function is used whenever different output formulas are needed for different inputs;
that is, more than one piece is needed to give a full description of what the function does!

Whenever you need one rule for one set of inputs, and another rule for a different set of inputs,
then a piecewise-defined function comes to the rescue.

As you'll observe in the definition below, the following notation is used for piecewise-defined functions:

• a single large left brace is used to gather together the different rules;
  there is no closing right brace
• to the right of this brace, each rule is given its own line:
  two ‘pieces’ result in two lines;
  three ‘pieces’ result in three lines, and so on
• each line has this pattern:
  • state the rule, followed by an optional comma;
  • followed by the word ‘ if ’ or the word ‘ for ’;
  • followed by the input(s) to which the rule applies;
  • followed by an optional comma (for all except the last line)

I tend to prefer the simplest notation that clearly conveys meaning, so I opted for no commas in the definition below.
However, some alternate presentations are offered following the definition.

Here are some alternate presentations of this piecewise-defined function:

$$ \cssId{s40}{ {|x|} = \begin{cases} x &\text{for }\ x\ge 0 \\ -x &\text{for }\ x\lt 0 \end{cases}} $$

$$ \cssId{s41}{ {|x|} = \begin{cases} x\,, &\text{if }\ x\ge 0 \\ -x\,, &\text{if }\ x\lt 0 \end{cases}} $$

$$ \cssId{s42}{ {|x|} = \begin{cases} x\,, &\text{if }\ x\ge 0\,, \\ -x\,, &\text{if }\ x\lt 0 \end{cases}} $$

$$ \cssId{s43}{ {|x|} = \begin{cases} x\,, &\text{for }\ x\ge 0\,, \\ -x\,, &\text{for }\ x\lt 0 \end{cases}} $$

• How to use the formula
  Here's how you'll use the formula to find $\,|x|\,$:
  Start by looking inside the absolute value, at the number $\,x\,$.
  • If $\,x\,$ is greater than or equal to zero, then use the first rule, $\,|x| = x\,$.
    That is, if a number is nonnegative, then its absolute value is itself.
    For example, $\,|3| = 3\,$.
  • If $\,x\,$ is less than zero, then use the second rule, $\,|x| = -x\,$.
    That is, if a number is negative, then its absolute value is its opposite.
    For example, $\,|-3| = -(-3) = 3\,$.
• The second line is most troublesome
  The second line of the rule is most troublesome:
  $|x| = -x\,, \text{ if }\ x \lt 0$
  One thing that leads to confusion is that people often read ‘$\,-x\,$’ as ‘negative $\,x\,$’,
  which makes it sound like the result is negative.
  NOT SO!!
  If $\,x\,$ is negative, then its opposite, $\,-x\,$, is positive!
  If you find yourself getting confused, just read ‘$\,-x\,$’ as ‘the opposite of $\,x\,$’,
  and this may clear up any difficulty.

The absolute value makes a useful appearance in the following formula:

So, what is $\,\sqrt{x^2}\,$?
There are two numbers which, when squared, give $\,x^2\,$:   $\,x\,$ and $\,-x\,$.
Which one do we want?
Answer:   Whichever one is nonnegative!
Unless we happen to know the sign of $\,x\,$, then we won't know which one is nonnegative.
Absolute value comes to the rescue!
The number $\,|x|\,$ has the same size (distance from zero) as both $\,x\,$ and $\,-x\,$, and is always nonnegative.

Don't ever fall into a common trap and think that $\,\sqrt{x^2}\,$ is just $\,x\,$.
Sure, this ‘formula’ works if we know, ahead of time, that $\,x\,$ is nonnegative.
But, usually we're in situations where $\,x\,$ is allowed to be any real number.
For example, let $\,x = -5\,$.
Then, $\,x^2 = (-5)^2 = 25\,$.
Is $\,\sqrt{x^2}\,$ equal to $\,x\,$ in this case?
No! $\,\sqrt{25}\ne -5\,$
However, it is true that $\, \cssId{s88}{\sqrt{x^2}} \cssId{s89}{= \sqrt{(-5)^2}} \cssId{s90}{= \overset{=\, |x|}{\overbrace{|-5|}}} \cssId{s91}{= 5}\,$.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
the Quadratic Formula