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Question 34071: Order after a Common Addend

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0067-34071 · q-20261005-advance-20261006-r01-order-relation-0096

Original question and givens

Under the usual real order, a=-22, b=-44, and c=8. Compare a+c and b+c, giving <, =, or > in that order.

  • G001 — Under the usual real order, a=-22, b=-44, and c=8. Compare a+c and b+c, giving <, =, or > in that order.

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-order-common-translation, [a, b, c, operation]]

After

derived = [common_translation, 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.

Step identifiers and dependencies

Rule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001

S002 · calculation

Before

Translation ORIGINAL a plus same c := sum(-22, 8)

After

Translation ORIGINAL a plus same c = -14

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Translation ORIGINAL b plus same c := sum(-44, 8)

After

Translation ORIGINAL b plus same c = -36

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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Translation ORIGINAL translated left minus right := sub(-14, -36)

After

Translation ORIGINAL translated left minus right = 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.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [-22, -44, 8]

After

derived = [-14, -36, 22, >]

Conditions: none

Adding the SAME c to BOTH sides preserves difference a−b.Subtract c from both to recover the original comparison,including equality and negative c.Never translate only one side;derived sums may exceed SAFE.

Step identifiers and dependencies

Rule: Order_COMMON_addend_cancel_BOTH_sides · Uses: G001, S004

Reverse checks

  • S005 · 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.
  • S004 · premise_and_result_audit — Recompute Translation ORIGINAL translated left minus right with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Translation ORIGINAL b plus same c with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Translation ORIGINAL a plus same c 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

>

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 4b07b5a222fe61aea45dd655ee0a2f2602778954ef5ce9b28d6a4fcbbe7a8829
  • candidate record SHA-256: 88f7b94663d036f2335f7c4d83f5fac0cc3a564c796b38cb765773803c56ad0c
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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