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Absolute Value Equation

An absolute value equation is an equation that contains an absolute value expression, such as ∣x−3∣=7|x - 3| = 7. To solve one, you typically split it into two separate equations — one for the positive case and one for the negative case — because the expression inside the absolute value bars could be either positive or negative.

An absolute value equation is an equation in which the variable appears inside absolute value notation. For an equation of the form ∣A∣=B|A| = B, where AA is an algebraic expression and B≥0B \geq 0, the solution is found by solving the two equations A=BA = B and A=−BA = -B. If B<0B < 0, the equation has no solution, since absolute value is always non-negative.

Key Formula

∣A∣=B  ⟹  A=BorA=−B(B≥0)|A| = B \implies A = B \quad \text{or} \quad A = -B \quad (B \geq 0)
Where:
  • AA = the algebraic expression inside the absolute value bars
  • BB = the non-negative number or expression that the absolute value equals

Worked Example

Problem: Solve ∣2x−5∣=9|2x - 5| = 9.
Step 1: Check that the right side is non-negative. Since 9≥09 \geq 0, the equation can have solutions.
Step 2: Write the two cases. The expression inside the absolute value is either equal to 9 or equal to −9-9.
2x−5=9or2x−5=−92x - 5 = 9 \quad \text{or} \quad 2x - 5 = -9
Step 3: Solve the first equation.
2x=14  ⟹  x=72x = 14 \implies x = 7
Step 4: Solve the second equation.
2x=−4  ⟹  x=−22x = -4 \implies x = -2
Step 5: Check both solutions in the original equation. ∣2(7)−5∣=∣9∣=9|2(7) - 5| = |9| = 9 ✓ and ∣2(−2)−5∣=∣−9∣=9|2(-2) - 5| = |-9| = 9 ✓.
Answer: The solutions are x=7x = 7 and x=−2x = -2.

Visualization

Why It Matters

Absolute value equations come up when you need to find values that are a certain distance from a point on a number line. In science and engineering, they appear in error tolerance problems — for instance, determining which measurements fall within an acceptable range of a target value. They also lay the groundwork for solving absolute value inequalities, which you'll encounter in later algebra courses.

Common Mistakes

Mistake: Forgetting to write two cases and only solving A=BA = B.
Correction: Absolute value strips away the sign, so the inside could be positive or negative. You must always set up both A=BA = B and A=−BA = -B to find all solutions.
Mistake: Trying to solve when the absolute value equals a negative number, like ∣x+1∣=−3|x + 1| = -3.
Correction: Absolute value can never be negative. If B<0B < 0, the equation has no solution. Recognizing this saves time and prevents phantom answers.

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