# Solving Simple Absolute Value Sentences

Need some simpler practice with absolute value first?

- Simplifying Basic Absolute Value Expressions
- Determining the Sign (positive or negative) of Absolute Value Expressions

Recall that
$\,|x|\,$ gives the distance between
$\,x\,$ and $\,0\,.$
If you think in terms of *distance*,
then it's easy to solve sentences involving absolute value!

## Example: An Absolute Value Equation

Note: We want all numbers $\,x\,$ whose distance from zero is $\,3\,.$

$$ \cssId{s16}{\overset{\text{whose distance from zero ...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$... }}{\ \strut x\ }|\ }} \overset{\text{is ...}}{\strut \ \ \ =\ \ \ } \overset{\text{three}}{\strut 3}} $$
The diagram above shows how the sentence
is *telling you* what you want!
Interpret it in the following order:

1) | We want numbers $\,x\,$... | the unknown is $\,x\,$ |

2) | whose distance from zero ... | the vertical bars, $|\ |\,,$ ask for distance from zero |

3) | is ... | the equal sign |

4) | $3$ | three |

Remember that you can ‘walk’ from zero in two directions: to the right, and to the left. The number $\,3\,$ is three units from zero to the right; the number $\,-3\,$ is three units from zero to the left.

The word ‘or’ in the sentence
‘$\,x=3\text{ or }x=-3\,$’ is the
*mathematical* word
‘or’.
You may want to review its meaning:
Practice with the Mathematical Words
‘and’,
‘or’,
‘is equivalent to’

## Example: An Absolute Value Inequality Involving ‘Less Than’

Note: We want all numbers $\,x\,$ whose distance from zero is less than $\,3\,.$

$$ \cssId{s40}{\overset{\text{whose distance from zero ...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$... }}{\ \strut x\ }|\ }} \overset{\text{is less than ...}}{\strut \ \ \ \ \ \lt\ \ \ \ \ } \overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ }} $$
The diagram above shows how the sentence
is *telling you* what you want!
Interpret it in the following order:

1) | We want numbers $\,x\,$... | the unknown is $\,x\,$ |

2) | whose distance from zero ... | the vertical bars, $|\ |$, ask for distance from zero |

3) | is less than ... | the ‘less than’ symbol |

4) | $3$ | three |

You can walk less than three units to the right—this gets you the numbers from $\,0\,$ to $\,3\,.$ You can walk less than three units to the left—this gets you the numbers from $\,0\,$ to $\,-3\,.$ Together, you end up with all the numbers between $\,-3\,$ and $\,3\,$:

The sentence ‘$\,-3 \lt x \lt 3\,$’
is just a *shorthand* for
‘$\,-3\lt x\,$
and
$\,\ x \lt 3\,$’.
That's the *mathematical* word
‘and’.
You may want to review its meaning:
Practice with the Mathematical Words
‘and’,
‘or’,
‘is equivalent to’

The shorthand ‘$\,-3\lt x\lt 3\,$’
is a *great* shorthand, because you see an
$\,x\,$ trapped between $\,-3\,$ and
$\,3\,$;
and those are precisely the values of $\,x\,$
that make the sentence true.

## Example: An Absolute Value Inequality Involving ‘Greater Than’

Note: We want all numbers $\,x\,$ whose distance from zero is greater than $\,3\,.$

$$ \cssId{s66}{\overset{\text{whose distance from zero ...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$... }}{\ \strut x\ }|\ }} \overset{\text{is greater than ...}}{\strut \ \ \ \ \ \gt\ \ \ \ \ } \overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ }} $$
The diagram above shows how the sentence is
*telling you* what you want!
Interpret it in the following order:

1) | We want numbers $\,x\,$... | the unknown is $\,x\,$ |

2) | whose distance from zero ... | the vertical bars, $|\ |$, ask for distance from zero |

3) | is greater than ... | the ‘greater than’ symbol |

4) | $3$ | three |

You can walk more than three units to the right—this gets you all the numbers to the right of $\,3\,.$ You can walk more than three units to the left—this gets you all the numbers to the left of $\,-3\,.$ Together, you end up with the two pieces shown below:

The word ‘or’ in the sentence
‘$\,x\lt -3\,$
or
$\, x\gt 3\,$’
is the
*mathematical* word
‘or’.
You may want to review its meaning:
Practice with the Mathematical Words
‘and’,
‘or’,
‘is equivalent to’

## Example: An Absolute Value Sentence With No Solutions

## Example: An Absolute Value Sentence That Is Always True

## Concept Practice

Solve the given absolute value sentence. Write the result in the most conventional way.

For more advanced students, a graph is available. For example, the equation $\,|x| = 3\,$ is optionally accompanied by the graph of $\,y = |x|\,$ (the left side of the equation, dashed green) and the graph of $\,y = 3\,$ (the right side of the equation, solid purple). In this example, you are finding the values of $\,x\,$ where the green graph intersects the purple graph.

Click the ‘Show/Hide Graph’ button to toggle the graph.