Question 4719: Original Rational-Expression Domain Guard

Original question and givens

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(17)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 21/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

• G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(17)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 21/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-variable-expression

Step-by-step derivation

typed rational expression and explicitly given environment

evaluation,not solving an equation

candidate0 original denominator := sub(1, 1)

candidate0 original denominator = 0

x=1;x−b=0

legal=No;E(x)=undefined

candidate1 original denominator := sub(21/2, 1)

candidate1 original denominator = 19/2

candidate1 E(x) := div(17, 19/2)

candidate1 E(x) = 34/19

x=21/2;x−b=19/2

legal=Yes;E(x)=34/19

candidate2 original denominator := sub(-19, 1)

candidate2 original denominator = -20

candidate2 E(x) := div(17, -20)

candidate2 E(x) = -17/20

x=-19;x−b=-20

legal=Yes;E(x)=-17/20

Reverse checks

• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute candidate2 E(x) with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute candidate2 original denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute candidate1 E(x) with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute candidate1 original denominator with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute candidate0 original denominator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([No, Yes, Yes], [undefined, 34/19, -17/20])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.