Algebra / Linear Inequalities
Absolute Value Inequalities
Absolute value inequalities describe distances. Inequalities of the form $|x-a|<b$ describe values within $b$ units of $a$, while $|x-a|>b$ describes values more than $b$ units away.
Least you need to know
- $|x-a|\le b$ means $a-b\le x\le a+b$ when $b\ge0$.
- $|x-a|<b$ means $a-b<x<a+b$ when $b>0$.
- $|x-a|>b$ means $x<a-b$ or $x>a+b$ when $b>0$.
- Count integer solutions only after translating the inequality.
Key notation
- $|x-4|\le2$ — Values of $x$ within 2 units of 4
Worked example
- $|x-4|\le2$ means $-2\le x-4\le2$.
- Add 4 to all parts: $2\le x\le6$.
- The integer solutions are $2,3,4,5,6$.
Common mistakes
- Less-than absolute value inequalities make a bounded interval.
- Greater-than absolute value inequalities split into two rays.
- Strict endpoints are not included.
How to recognize it
- The problem asks how many integer values satisfy an absolute value inequality.
- The inequality contains $|x-a|$.
- The wording describes distance less than, at most, greater than, or at least a number.
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