Math101 · Independent Ontario learning library
Math101Absolute Value Inequalities
Absolute value inequalities describe distance ranges. For $k>0$, $|u|<k$ means $-k<u<k$ (inside a band), while $|u|>k$ means $u<-k$ or $u>k$ (outside the band).
These inequalities model allowable error, measurement tolerance, and exclusion zones. Their geometry makes compound inequalities and set unions meaningful.
Intuition and core definition
Absolute value inequalities describe distance ranges. For $k>0$, $|u|<k$ means $-k<u<k$ (inside a band), while $|u|>k$ means $u<-k$ or $u>k$ (outside the band). Inclusive symbols produce inclusive endpoints. Negative or zero bounds require separate logical analysis.
Notation, language, and conditions
$|x-a|\le k$ describes the closed interval $[a-k,a+k]$ for $k\ge0$. $|x-a|>k$ describes two rays $(-\infty,a-k)\cup(a+k,\infty)$. “And” corresponds to intersection between bounds; “or” corresponds to the union of outside regions.
Why this idea matters
Absolute-value inequalities describe points inside or outside a distance band, connecting compound inequalities to geometric intervals.
A dependable method
- Isolate the absolute-value expression and inspect the bound.
- Translate a “less than” comparison into a compound AND inequality.
- Translate a “greater than” comparison into two OR inequalities.
- Solve each part, preserving endpoint inclusion.
- Graph and test a centre point plus points inside and outside the boundaries.
