70.4 Relation Transformation Errors

Relation Transformation Errors are the category of mistakes arising when an equation or inequality is manipulated from one form into another, covering broken equation balance, mishandled inequality direction, omitted zero-product conditions, lost solution branches, unchecked risky transformations, and skipped confirmation that a transformation preserved the original relation.

Equation Balance Broken

Recognizing When Both Sides Were Not Treated Identically

This diagnosis identifies a step in which an operation was applied to only one side of an equation, or applied differently to each side, breaking the balance the equation depended on.

Why This Diagnosis Compares Both Sides' Treatment Directly

Because the validity of this kind of step depends specifically on both sides receiving the identical operation, this diagnosis directly compares what was actually done to the left side against what was actually done to the right side at that specific step.

Inequality Direction Mishandled

Recognizing When the Inequality Symbol Was Not Reversed When Required

This diagnosis identifies a step in which both sides of an inequality were multiplied or divided by a negative value without reversing the direction of the inequality symbol as that operation requires.

Why This Diagnosis Checks the Sign of the Multiplier or Divisor Specifically

Because the direction-reversal rule applies specifically and only when the value involved is negative, this diagnosis checks the sign of that specific value at the step in question, rather than checking the general type of operation alone.

Zero-Product Condition Omitted

Recognizing When a Factored Equation Was Not Set Equal to Zero First

This diagnosis identifies a step in which the zero-product principle was applied to a factored expression before that expression was actually confirmed to equal zero.

Why This Diagnosis Confirms the Zero-Side Condition First

Because the zero-product principle depends entirely on the product actually equaling zero, this diagnosis confirms that this specific condition was genuinely met before checking whether the principle itself was applied correctly.

Solution Branch Lost

Recognizing When One of Several Valid Branches Was Dropped

This diagnosis identifies a point in the work where a solving process that should have produced more than one branch, such as a plus-or-minus square root or multiple factors, was followed through for only one of those branches.

Why This Diagnosis Counts the Expected Branches Directly

Because the number of branches a given technique should produce is generally known in advance, this diagnosis directly counts how many branches actually appear in the work and compares that count against the number the technique should have produced.

Nonreversible Transformation Used

Recognizing an Unflagged Risky Transformation

This diagnosis identifies a step involving a transformation, such as squaring both sides or clearing a denominator, that carries a known risk of introducing an extraneous candidate, without that risk having been flagged or accounted for later in the work.

Why This Diagnosis Draws Directly on Established Risk Categories

This diagnosis applies the same specific categories of risky transformation already established during transformation risk control, checking the work for the presence of one of those specific categories at the step in question.

Original Relation Check Omitted

Recognizing When a Final Candidate Was Never Verified

This diagnosis identifies work that concludes with a candidate value but shows no evidence that this candidate was ever substituted back into the original relation to confirm it.

Why This Diagnosis Looks for Explicit Verification Evidence

Because verification is a distinct, visible step when performed properly, this diagnosis looks specifically for that explicit evidence in the work, rather than assuming verification occurred simply because a final answer was reached.

Equivalent-Step Confirmation

Confirming a Transformation Genuinely Preserved the Original Relation

This diagnosis directly tests whether a specific transformation applied during the work actually produced an equivalent relation, by checking whether a value known to satisfy the earlier line still satisfies the later line.

Why This Confirmation Serves as a General-Purpose Check

Because this test can be applied to any type of transformation, regardless of which specific category it falls into, it serves as a general-purpose confirmation method usable even when the specific transformation involved does not match one of the other, more specific error categories described here.