SOLVING SIMPLE ABSOLUTE VALUE SENTENCES

Recall that [beautiful math coming... please be patient] $\,|x|\,$ 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
Solve: [beautiful math coming... please be patient] $|x| = 3$
Answer:
[beautiful math coming... please be patient] $x = 3\ \text{ or }\ x=-3$
Note:
We want all numbers $\,x\,$ whose distance from zero is $\,3\,$.
[beautiful math coming... please be patient] $$ \overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }} \overset{\text{is...}}{\strut \ \ \ =\ \ \ } \overset{\text{three}}{\strut 3} $$ The diagram above shows how the sentence is telling you what you want!
Interpret it in the following order:
1) We want numbers [beautiful math coming... please be patient] $\,x\,$... (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 [beautiful math coming... please be patient] $\,3\,$ is three units from zero to the right;
the number $\,-3\,$ is three units from zero to the left.

solving an absolute value equation

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’
Solve: [beautiful math coming... please be patient] $|x| \lt 3$
Answer:
[beautiful math coming... please be patient] $-3 \lt x \lt 3$
Note:
We want all numbers $\,x\,$ whose distance from zero is less than $\,3\,$.
[beautiful math coming... please be patient] $$ \overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }} \overset{\text{is less than...}}{\strut \ \ \ \ \ \lt\ \ \ \ \ } \overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ } $$ The diagram above shows how the sentence is telling you what you want!
Interpret it in the following order:
1) We want numbers [beautiful math coming... please be patient] $\,x\,$... (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 [beautiful math coming... please be patient] $\,0\,$ 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 $\,3\,$:

solving an absolute value inequality

The sentence ‘$\,-3 \lt x \lt 3\,$’ is just a shorthand for ‘$\,-3\lt x\ \text{ and }\ x<3\,$’.
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 [beautiful math coming... please be patient] $\,x\,$ 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’
Solve: [beautiful math coming... please be patient] $|x| \gt 3$
Answer:
[beautiful math coming... please be patient] $x \lt -3\ \text{ or }\ x\gt 3$
Note:
We want all numbers $\,x\,$ whose distance from zero is greater than $\,3\,$.
[beautiful math coming... please be patient] $$ \overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }} \overset{\text{is greater than...}}{\strut \ \ \ \ \ \gt\ \ \ \ \ } \overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ } $$ The diagram above shows how the sentence is telling you what you want!
Interpret it in the following order:
1) We want numbers [beautiful math coming... please be patient] $\,x\,$... (the unknown is $\,x\,$)
2) whose distance from zero... (the vertical bars, [beautiful math coming... please be patient] $|\ |$, 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:

solving an absolute value inequality

The word ‘or’ in the sentence ‘$\,x\lt -3\ \text{ or }\ x\gt 3\,$’ 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
Solve: [beautiful math coming... please be patient] $|x| \lt -1$
Solution:
There are no solutions, since distance can't be negative.
That is, [beautiful math coming... please be patient] $\,|x|\,$ is always greater than or equal to zero; so, it can't ever be less than $\,-1\,$.
EXAMPLE: an absolute value sentence that is always true
Solve: [beautiful math coming... please be patient] $|x| \gt -1$
Solution:
All real numbers are solutions, since all distances are nonnegative.
That is, [beautiful math coming... please be patient] $\,|x|\,$ is always greater than or equal to zero; so, it must also be greater than $\,-1\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Solving Sentences Involving ‘Plus or Minus’

 
 
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

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

For more advanced students, a graph is displayed.
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 if you prefer not to see the graph.

Solve: