Formulas & tables/Mathematics/Absolute value/Inequalities with absolute value
Inequalities with absolute value

Inequalities with absolute value

∣x−a∣≤r  ⟺  a−r≤x≤a+r|x-a|\le r \iff a-r\le x\le a+r

Result unit: — (interval)

What it describes

An inequality with an absolute value defines an interval around aa with radius rr.

Symbols

xx
variable
aa
center of the interval
rr
radius of the interval

Where it comes from

∣x−a∣≤r|x-a|\le r means "xx is at most rr away from aa", i.e. it lies in the interval [a−r, a+r][a-r,\,a+r]. This is the geometric interpretation of the modulus as a distance.

When to use it

Solving inequalities with absolute values, intervals.

Example

Solve the inequality ∣x−2∣≤3|x-2|\le3.
Substitutiona=2a=2, r=3r=3.
Result−1≤x≤5-1\le x\le5.

Related formulas

Definition of absolute value Properties of absolute value

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