Absolute Value Equations
The absolute value of a real number x, written |x|, measures its distance from zero on the number line — distance is always non-negative. For example, |7| = 7 and |−7| = 7, because both 7 and −7 are exactly 7 units from zero. This geometric meaning is the key to solving absolute value equations: when we write |expression| = c, we are asking which values of the variable place the expression exactly c units from zero. Absolute value equations are classified as non-linear because the absolute value function has a 'corner' at the origin and cannot be written as a single straight-line rule — it behaves differently on either side of zero. This means a single absolute value equation typically splits into two linear (or sometimes quadratic) cases, and we must check every proposed solution in the original equation to rule out extraneous roots introduced during algebra. Mastering absolute value equations builds essential problem-solving habits: identifying cases, solving each branch, and verifying answers. These skills recur in calculus (the formal definition of a limit uses absolute value), optimization, error analysis in science, and programming (the built-in abs() function appears in countless algorithms). Understanding the structure of |f(x)| = g(x) is therefore a gateway concept in algebra and analysis.
- Isolate the absolute value expression
- Split into two cases
- Solve each case independently
- Check for extraneous solutions
- Write the solution set
Isolate the absolute value expression
Before splitting into cases, rearrange the equation so the absolute value stands alone on one side: |f(x)| = c. If c < 0, stop immediately — an absolute value can never equal a negative number, so the equation has no solution. If c = 0, the only solution is f(x) = 0 (no splitting needed). If c > 0, proceed to step 2.
Split into two cases
The definition |u| = c (c > 0) means u = c OR u = −c. Apply this to both sides: write Case 1: f(x) = c and Case 2: f(x) = −c. These two equations correspond to the two branches of the absolute value function — the expression inside is either positive (or zero) or negative.
Solve each case independently
Use standard algebraic techniques (inverse operations, distribution, combining like terms) to solve each of the two equations separately. Each case may yield one solution, no solution, or infinitely many solutions depending on the complexity of f(x).
Check for extraneous solutions
Substitute every candidate solution back into the ORIGINAL absolute value equation. A solution is valid only if both sides are equal. Extraneous solutions sometimes appear when the algebra has manipulated an equation in a way that introduced values inconsistent with the original statement. Always verify: skip this step and you risk reporting wrong answers.
Write the solution set
List all solutions that passed the check. Express the solution set in set notation, e.g., {−3, 5}, or state 'No solution' (∅) if no values work. For equations with infinitely many solutions, describe them with interval or set-builder notation.
The Key Equation
- |f(x)| — Absolute value of f(x) (same units as x): The distance of f(x) from zero; always non-negative.
- f(x) — Argument expression (varies): Any algebraic expression in the variable x; e.g., 2x − 3 or x² − 5.
- c — Constant (right-hand side) (dimensionless): The value the absolute value is set equal to. Must satisfy c ≥ 0 for real solutions to exist.
Worked Example
Problem. Solve: |3x − 6| + 2 = 11
Answer. x = 5 or x = −1
Origins · 1 minKarl Weierstrass and how this was discovered
Weierstrass gave the first fully rigorous definition of the limit of a function — the famous epsilon-delta (ε-δ) definition — in which absolute value is used to express 'closeness': |f(x) − L| < ε whenever 0 < |x − a| < δ. This formalism placed the entire edifice of calculus on a solid foundation and made absolute value a central notation in analysis. He also pioneered the theory of complex functions and is credited with constructing a continuous nowhere-differentiable function, shocking the mathematical world. Weierstrass did not publish much until his 40s — most of his revolutionary ideas circulated through letters and lecture notes — yet he became one of the most influential mathematicians of the 19th century, earning the title 'father of modern analysis.'
Related Concepts
- Linear Equations — Each branch of an absolute value equation is a linear equation; strong linear-equation skills are a prerequisite.
- Absolute Value Inequalities — The same split-case logic applies to |f(x)| < c and |f(x)| > c, extending the concept to solution intervals.
- Piecewise Functions — Absolute value is itself a piecewise function: |x| = x for x ≥ 0 and |x| = −x for x < 0.
- Quadratic Equations — Equations like |x²− 4| = 5 combine absolute value with quadratics, requiring both skill sets.