28.3 Absolute Value Equation Resolution

Absolute Value Equation Resolution is the complete procedure for solving an absolute value equation whose enclosed expression is a full linear expression rather than a bare variable, applying the dual-branch structure established in Absolute Value Equation Structure to a Linear Enclosed Expression that may itself require multi-step solving once each branch has been formed.

Enclosed Expression Equal to the Positive Radius is the formation of the first branch equation, setting the entire linear expression found within the absolute value bars, exactly as written, equal to the positive numerical value on the opposite side of the original equation, referred to here as the radius by analogy with the exact distance such a value represents from zero, shown in the general form below.

where c is the positive radius. This branch is formed by simply removing the absolute value bars and retaining the enclosed expression and the radius value exactly as they appear.

Enclosed Expression Equal to the Negative Radius is the formation of the second branch equation, setting the identical linear expression found within the absolute value bars equal to the negative of that same radius value, shown in the general form below.

This branch accounts for the case in which the enclosed linear expression itself evaluates to a negative number whose absolute value nonetheless equals the stated positive radius.

Independent Positive-Branch Resolution is the action of solving the equation formed through Enclosed Expression Equal to the Positive Radius entirely on its own, using whichever standard technique applies, whether One-Step Solving Procedure or Multi-Step Solving Sequence, without reference to the second branch, producing one candidate solution for the variable.

Independent Negative-Branch Resolution is the corresponding action of solving the equation formed through Enclosed Expression Equal to the Negative Radius entirely on its own, again using whichever standard technique applies, producing a second candidate solution for the variable, obtained completely independently of the first branch's solving process.

Distinct Solution Collection is the action of gathering the two candidate solutions produced by Independent Positive-Branch Resolution and Independent Negative-Branch Resolution into a single reported collection, recognizing that these two values are ordinarily different from one another, though in special cases they may coincide, and that both are candidates requiring confirmation before being finalized as the Absolute Value Equation Solution Set.

Original Absolute Equation Verification is the concluding action, substituting each candidate solution from Distinct Solution Collection back into the original equation exactly as it was first presented, with the absolute value bars still intact, evaluating the enclosed linear expression at that candidate value, applying the absolute value operation to the result following Absolute Value after Numerical Simplification, and confirming that this final nonnegative value matches the radius originally given. A candidate that fails this verification, an outcome that can arise if an earlier step introduced an error, must be discarded rather than reported as part of the final solution set.