Algebra / Linear Equations One Variable
Absolute Value Equations and Distance
Absolute value equations can be interpreted as distance from zero or from a center point. Equations of the form $|x-a|=b$ usually have two solutions when $b>0$.
Least you need to know
- $|x-a|=b$ means $x$ is $b$ units from $a$.
- If $b>0$, $|x-a|=b$ has two solutions.
- If $b=0$, $|x-a|=0$ has one solution.
- If $b<0$, $|x-a|=b$ has no solution.
Key notation
- $|x-5|=3$ — The distance from $x$ to 5 is 3
Worked example
- To solve $|x-5|=3$, set $x-5=3$ or $x-5=-3$.
- The solutions are $x=8$ and $x=2$.
Common mistakes
- Absolute value cannot be negative.
- An absolute value equation can have two solutions.
- The center point is the value that makes the expression inside the absolute value equal to zero.
How to recognize it
- The equation contains absolute value bars.
- The problem describes distance on a number line.
- The problem asks for a lesser, greater, or number of solutions.
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