69.3 Candidate Value Verification

Candidate Value Verification is the systematic process of testing every candidate value produced by a solving process against the original equation or inequality, deciding for each one whether it is genuinely accepted as a solution or formally rejected, and keeping a clear record of that decision.

Candidate Value Collection

Gathering Every Produced Candidate

Verification begins by gathering every candidate value produced during the solving process into a single collected list, before any individual candidate is tested.

Why Collection Happens before Testing

Collecting every candidate first, rather than testing each one as soon as it is found, ensures that the full scope of candidates to be verified is known in advance, reducing the risk that one is overlooked partway through the process.

Original-Relation Candidate Substitution

Substituting Each Candidate into the Original Relation

Each collected candidate value is substituted, one at a time, into the original equation or inequality exactly as it was first presented.

Why the Original Form Is Used for This Substitution

Using the original, unmodified relation, rather than any intermediate equation produced during solving, ensures the test genuinely confirms whether the candidate satisfies the statement that was actually meant to be solved.

Undefined Candidate Rejection

Rejecting Candidates That Make the Relation Undefined

If substituting a candidate value causes any part of the original relation to become undefined, such as producing a zero denominator, that candidate is immediately rejected without further evaluation.

Why This Rejection Happens before Any Further Evaluation

Because an undefined expression cannot be meaningfully evaluated as true or false, this rejection must occur immediately upon discovering the undefined result, rather than attempting to continue evaluating a relation that no longer has a meaningful value.

Equation Truth Evaluation

Evaluating Whether an Equation Holds True

For a candidate that does not produce an undefined result, both sides of the substituted equation are simplified and compared, determining whether they are equal.

Why This Evaluation Directly Determines Acceptance

Because an equation is satisfied precisely when both of its sides are equal, this direct comparison is the exact test that determines whether the candidate is accepted as a genuine solution to that equation.

Inequality Membership Evaluation

Evaluating Whether an Inequality Holds True

For a candidate associated with an inequality rather than an equation, the substituted inequality is evaluated to determine whether the resulting comparison is true.

Why This Evaluation Differs from Equation Testing

Unlike an equation, which requires exact equality, this evaluation tests whether the specific inequality relationship holds true for the substituted values, matching the different kind of condition an inequality expresses compared to an equation.

Candidate Acceptance Decision

Deciding to Accept a Candidate

A candidate value is formally accepted once it has passed both the undefined-result check and the truth or membership evaluation, confirming it as a genuine solution.

Why Acceptance Requires Passing Both Checks

Requiring a candidate to pass both checks ensures that only values which are both mathematically defined and genuinely satisfy the original relation are accepted, rather than accepting a value based on only a partial confirmation.

Candidate Rejection Record

Recording Every Rejected Candidate and the Reason

Each candidate that fails verification, whether due to being undefined or failing the truth evaluation, is recorded along with the specific reason for its rejection.

Why Recording the Reason for Rejection Matters

Keeping a specific reason alongside each rejected candidate makes it possible to review the solving process afterward and understand exactly why a particular value did not qualify as a genuine solution, rather than leaving only an unexplained omission.