Advanced Math / Absolute Value Equations
Absolute value as distance and solution counts
Absolute value can model distance on a number line. This viewpoint makes solution counts and tolerance statements easier to interpret.
Least you need to know
- Distance is always nonnegative.
- A positive distance from a point gives two locations.
- A distance of zero gives one location.
Key notation
- |x-a| — The distance between $x$ and $a$
Worked example
The statement 'x is within 3 units of 10' is $|x-10|\le 3$.
Common mistakes
- Distance from $a$ is written $|x-a|$, not $|x+a|$.
- Within means at most, not at least.
How to recognize it
- The wording includes distance, within, tolerance, or number of solutions.
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