Advanced Math / Absolute Value Inequalities
Absolute value inequalities
Absolute value inequalities describe distances. Less-than inequalities produce inside intervals; greater-than inequalities produce outside intervals.
Least you need to know
- $|x-a|<r$ means $a-r<x<a+r$.
- $|x-a|>r$ means $x<a-r$ or $x>a+r$.
- Endpoint inclusion follows the original inequality symbol.
Key notation
- r — A positive radius or distance from a center
Worked example
For $|x-4|\le 6$, the values of $x$ are within 6 of 4, so $-2\le x\le 10$.
Common mistakes
- Less than uses and; greater than uses or.
- Strict endpoints are not counted in integer counts.
How to recognize it
- The problem asks for an interval or number of integer values.
- The stem uses words like within, farther than, at most, or more than.
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