Mathematics.

equations

Absolute Value Equations

Algebra I20 minDifficulty3 out of 10

You should know: absolute value, linear equation

Overview

An absolute value equation contains a variable expression inside absolute value bars, such as |x - 3| = 7. Because |a| represents distance from zero, an absolute value equation typically splits into two separate linear equations — one for the case where the inside expression is positive, and one where it's negative — each of which must be solved separately.

Interactive Graph

Graph of |x-3| and where it crosses a value

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Formal Definition

Definition

For c ≥ 0, the equation |A| = c splits into two cases:

∣A∣=c (c≥0)  ⟺  A=c or A=−c|A| = c \ (c\ge 0) \iff A=c \text{ or } A=-c
Case split

Properties

No solution case

∣A∣=c with c<0 has no solution (absolute value is never negative)|A|=c \text{ with } c<0 \text{ has no solution (absolute value is never negative)}

Isolate first

Always isolate the absolute value expression before splitting into cases\text{Always isolate the absolute value expression before splitting into cases}

Worked Examples

  1. Split into two cases.

    2x−1=9 or 2x−1=−92x-1=9 \text{ or } 2x-1=-9

  2. Solve each case separately.

    2x=10⇒x=5or2x=−8⇒x=−42x=10\Rightarrow x=5 \quad\text{or}\quad 2x=-8\Rightarrow x=-4

Answer: x = 5 or x = -4

Practice Problems

Difficulty 3/10

Solve: |3x + 4| - 2 = 7.

Common Mistakes

Common Mistake

Splitting into two cases before isolating the absolute value expression, e.g. treating |2x-1|+3=9 as 2x-1=9 or 2x-1=-9.

Always isolate the absolute value expression FIRST (subtract the 3, giving |2x-1|=6) before splitting into the positive and negative cases.

Common Mistake

Not checking whether the isolated constant is negative, and still attempting to split into two cases.

If, after isolating, the equation reads |A| = c where c < 0, the equation has NO SOLUTION — absolute value can never produce a negative output.

Summary

  • |A| = c (c ≥ 0) splits into A = c or A = -c.
  • Always isolate the absolute value expression before splitting into cases.
  • If the isolated constant is negative, the equation has no solution.

References

  1. WebsiteWikipedia — Absolute value