Under the usual real order, is (1)/2 in the closed interval [-9, 11]? The denominator is positive and both endpoints are included.
G001 — Under the usual real order, is (1)/2 in the closed interval [-9, 11]? The denominator is positive and both endpoints are included.
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.
source roles = [qf-order-closed-interval, [lower, operation, p, q, upper]]
After
derived = [closed_interval, no-invented-domain]
Conditions: none
Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.
Interval ORIGINAL lower times positive denominator := mul(-9, 2)
After
Interval ORIGINAL lower times positive denominator = -18
Conditions: none
Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.
Interval ORIGINAL upper times positive denominator := mul(11, 2)
After
Interval ORIGINAL upper times positive denominator = 22
Conditions: none
Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.
For a positive denominator,lower≤p/q≤upper is equivalent to lower*q≤p≤upper*q.BOTH endpoints are included.A singleton is valid but contains only its endpoint.Do not use open inequalities or silently sort reversed endpoints.
S004 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
S003 · premise_and_result_audit — Recompute Interval ORIGINAL upper times positive denominator with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Interval ORIGINAL lower times positive denominator with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
Result
Yes
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.