Math101 · Independent Ontario learning library
Math101Absolute Value Equations
An absolute value equation describes a distance condition. If $|u|=k$ with $k>0$, then $u=k$ or $u=-k$ because two points lie distance $k$ from zero.
Distance equations model tolerances, deviations, and symmetric locations. The sign cases teach careful logical branching used in many algebraic equations.
Intuition and core definition
An absolute value equation describes a distance condition. If $|u|=k$ with $k>0$, then $u=k$ or $u=-k$ because two points lie distance $k$ from zero. If $k=0$, only $u=0$; if $k<0$, there is no real solution. These cases must be checked before splitting.
Notation, language, and conditions
For $|x-a|=k$, solutions lie $k$ units from centre $a$: $x=a\pm k$. The bars must be isolated before applying the two-case rule. An equation containing extra terms, such as $2|x-1|+3=11$, is first reduced to $|x-1|=4$.
Why this idea matters
An absolute-value equation translates a fixed distance into two possible directions, producing two branches unless the target distance is zero.
A dependable method
- Isolate the absolute-value expression.
- Inspect the other side: negative means no solution, zero means one central solution, positive means two cases.
- For a positive value, write the inside equal to both the positive and negative target.
- Solve each resulting equation.
- Substitute every candidate into the original, especially when other algebraic operations are present.
