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.

下一节推荐课程

通过 Compound inequalities and integer counts 继续学习此主题。

Compound inequalities and integer counts