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 $\,-1\,$.
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.