Absolute Value Equation Solver

In algebra, the absolute value of a real number x, denoted |x|, is formally defined as:

Geometrically, |x| represents the Euclidean distance between the point x and the origin (0) along the 1D real number line. Because distance is inherently a scalar geometric magnitude, absolute value is strictly non-negative: |x| ≥ 0 for all real x.

When generalized to a linear binomial expression |x − x₀| = r, the equation states that the distance between a variable point x and a fixed reference center point x₀ is exactly equal to the radius r. This immediately establishes why there are two geometric solutions:

• Rightward branch: x = x₀ + r (stepping r units to the right of the pivot center).
• Leftward branch: x = x₀ − r (stepping r units to the left of the pivot center).

For an equation in standard form |ax + b| = c with a ≠ 0, factoring out |a| yields |x − (−b/a)| = c / |a|. Here, the geometric center point is x₀ = −b/a, and the radius distance is r = c / |a|.

To solve any absolute value equation algebraically, follow this systematic multi-step protocol:

1. Step 1: Isolate the Absolute Value Term: If the equation has outside operations (e.g. 3|2x − 1| + 5 = 17), subtract 5 and divide by 3 first: |2x − 1| = 4. Never attempt to branch before the bars are isolated on one side.
2. Step 2: Inspect the Right-Hand Side (RHS):
   • If RHS < 0, the equation has no real solutions (S = ∅).
   • If RHS = 0, the equation has exactly one solution: ax + b = 0 ⇒ x = −b/a (the vertex of the absolute value function).
   • If RHS > 0, proceed to dual branch splitting.
3. Step 3: Split into Positive and Negative Branches:
   • Branch 1 (Positive): ax + b = +c
   • Branch 2 (Negative): ax + b = −c
4. Step 4: Solve Each Linear Equation: Isolate x algebraically in both branches:
   x₁ = (c − b) / a   and   x₂ = (−c − b) / a
5. Step 5: Verify Candidate Roots: Substitute each root back into the original equation to ensure both sides evaluate to identical numerical values.

When the variable appears both inside and outside the absolute value bars, such as in equations of the form |ax + b| = cx + d, solving the equation can produce false mathematical artifacts known as extraneous solutions.

An extraneous solution is a root derived logically through valid algebraic manipulations that nevertheless fails to satisfy the original equation. In absolute value equations, extraneous roots occur because the algebraic branching operation:

implicitly assumes that the right-hand side is non-negative: cx + d ≥ 0. If solving one of these linear branches produces a root x* where c(x*) + d < 0, then the left side evaluates to a positive magnitude while the right side evaluates to a negative number. Because |U| ≥ 0 is impossible to equal a negative number, x* is extraneous and must be discarded.

When both sides of the equation are enclosed in absolute value bars:

Students often wonder if four separate cases are required: (+, +), (+, −), (−, +), and (−, −). Mathematically, four cases are redundant because:

• (+, +) is identical to (−, −) upon multiplying both sides by −1.
• (+, −) is identical to (−, +) upon multiplying both sides by −1.

Therefore, exactly two linear equations suffice:

1. Case 1: ax + b = +(cx + d)
2. Case 2: ax + b = −(cx + d)

Crucially, equations of this form never yield extraneous solutions because both sides are intrinsically absolute values and thus guaranteed to be non-negative (|LHS| ≥ 0 and |RHS| ≥ 0).