Need some simpler practice with absolute value and related concepts first?

• Simplifying Basic Absolute Value Expressions
• Solving Simple Absolute Value Sentences
• Solving Sentences involving ‘Plus or Minus’

Now, let's talk about the concepts involved in solving absolute value equations:

Translating the Theorem

Recall first that normal mathematical conventions dictate that   ‘$\,|x| = k\ $’  represents an entire class of sentences, including the members

$|x| = 2$

$|x| = 5.7$

$|x| = \frac{1}{3}$

The variable $\,k\ $ changes from sentence to sentence, but is constant within a given sentence.

Also recall that  ‘$\,x=\pm k\ $’  is a shorthand for  ‘$\,x = k\,$ or $\,x = -k\,$’.

When you see a sentence of the form $\,|x| = k\ $, here’s what you should do:

• Check that $\,k\,$ is a nonnegative number (zero, or greater than zero).
• The symbol $\,|x|\,$ represents the distance between $\,x\,$ and $\,0\,.$
• Thus, you want the numbers $\,x\,,$ whose distance from $\,0\,$ is $\,k\,.$
• You can walk from $\,0\,$ in two directions: to the right, or to the left. Walk to the right a distance $\,k\,,$ and you get to the number $\,k\,.$ Walk to the left a distance $\,k\,,$ and you get to the number $\,-k\,.$
• Thus,  $\,|x| = k\ $  is equivalent to  $\,x = k\ \text{ or }\ x = -k\,,$ which goes by the shorthand  $\,x=\pm k\,.$
• Equivalent sentences are completely interchangeable, and you can use whichever is easiest to work with. In this case, you’re getting rid of the troublesome absolute value in exchange for a less-troublesome ‘plus or minus’ sign.

Recall that   ‘$\iff$’   is a symbol for ‘is equivalent to’.

The power of the sentence-transforming tool

goes far beyond solving simple sentences like $\,|x| = 5\,$!

Since $\,x\,$ can be any real number, you should think of $\,x\,$ as merely representing the stuff inside the absolute value symbols. Thus, you could think of rewriting the tool as:

See how this idea is used in the following examples:

Example

It’s a good idea to check your solutions:

$|2 - 3(-\frac{5}{3})|\ \overset{\text{?}}{=}\ 7$
$|2 + 5| = 7$
Check!

$|2 - 3(3)|\ \overset{\text{?}}{=}\ 7$
$|2 - 9| = 7$
Check!

Example

Example