Recall that
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$\,|x|\,$ 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:
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$|x| = 3$
Answer:
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$x = 3\ \text{ or }\ x=-3$
Note:
We want all numbers $\,x\,$ whose distance from zero is $\,3\,$.
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$$
\overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }}
\overset{\text{is...}}{\strut \ \ \ =\ \ \ }
\overset{\text{three}}{\strut 3}
$$
The diagram above shows how the sentence is
telling you what you want!
Interpret it in the following order:
1) |
We want numbers
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$\,x\,$... |
(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
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$\,3\,$ 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’
Solve:
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$|x| \lt 3$
Answer:
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$-3 \lt x \lt 3$
Note:
We want all numbers $\,x\,$ whose distance from zero is less than $\,3\,$.
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$$
\overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }}
\overset{\text{is less than...}}{\strut \ \ \ \ \ \lt\ \ \ \ \ }
\overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ }
$$
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
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$\,0\,$ 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 ‘$\,-3\lt x\ \text{ and }\ x<3\,$’.
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
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$\,x\,$ 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:
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$|x| \gt 3$
Answer:
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$x \lt -3\ \text{ or }\ x\gt 3$
Note:
We want all numbers $\,x\,$ whose distance from zero is greater than $\,3\,$.
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$$
\overset{\text{whose distance from zero...}}{\overbrace{\ |\overset{\text{want numbers $\,x\,$...}}{\ \strut x\ }|\ }}
\overset{\text{is greater than...}}{\strut \ \ \ \ \ \gt\ \ \ \ \ }
\overset{\text{three}}{\ \ \ \ \strut 3\ \ \ \ }
$$
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,
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$|\ |$, 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 ‘$\,x\lt -3\ \text{ or }\ x\gt 3\,$’ 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:
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$|x| \lt -1$
Solution:
There are no solutions, since distance can't be negative.
That is,
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$\,|x|\,$ 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:
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$|x| \gt -1$
Solution:
All real numbers are solutions, since all distances are nonnegative.
That is,
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$\,|x|\,$ is always greater than or equal to zero; so, it must also be greater than $\,-1\,$.
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.