Explore related knowledge and questions →
← Question derivation index · Original question batch
R0008-04715 · q-20261008-advance-20261011-r03-variable-expression-0090
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(13)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 19/2, -14/1]. 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)=(13)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 19/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).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.
Before
typed rational expression and explicitly given environment
After
evaluation,not solving an equation
Conditions: none
Evaluating an expression is not solving an equation;substitute only the supplied environment without inventing roots or changing bindings.
Rule: expression_environment_scope · Uses: G001
Before
candidate0 original denominator := sub(2, 2)
After
candidate0 original denominator = 0
Conditions: none
Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
x=2;x−b=0
After
legal=No;E(x)=undefined
Conditions: none
Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined.
Rule: original_expression_domain_guard · Uses: G001, S002
Before
candidate1 original denominator := sub(19/2, 2)
After
candidate1 original denominator = 15/2
Conditions: none
Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
candidate1 E(x) := div(13, 15/2)
After
candidate1 E(x) = 26/15
Conditions: none
Evaluate the original expression only with a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
x=19/2;x−b=15/2
After
legal=Yes;E(x)=26/15
Conditions: none
Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined.
Rule: original_expression_domain_guard · Uses: G001, S005
Before
candidate2 original denominator := sub(-14, 2)
After
candidate2 original denominator = -16
Conditions: none
Evaluate the original denominator x−b first;do not cancel a hole just because the numerator is0.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
candidate2 E(x) := div(13, -16)
After
candidate2 E(x) = -13/16
Conditions: none
Evaluate the original expression only with a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
x=-14;x−b=-16
After
legal=Yes;E(x)=-13/16
Conditions: none
Eligibility comes from the original expression's domain;even a=0 does not legalize the excluded point,and its value remains undefined.
Rule: original_expression_domain_guard · Uses: G001, S008
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.([No, Yes, Yes], [undefined, 26/15, -13/16])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 4d8b3d5ffdcb8c17c723d8265d757ae934168ab1604a0aee8d3092366c189de1candidate record SHA-256: 23ff03955369063641df86be113cfec8260d7c95fb6f86dbad52750e539633d3semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED