Question 10658: Inclusive Membership in a Declared Closed Rectangle

Original question and givens

Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10] and point [x,y]=[-3, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.

• G001 — Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10] and point [x,y]=[-3, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.

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-multivariable-real-function

Step-by-step derivation

rectangle coordinate0 lower slack := sub(-3, -9)

rectangle coordinate0 lower slack = 6

rectangle coordinate0 upper slack := sub(9, -3)

rectangle coordinate0 upper slack = 12

coordinate index=0;value=-3;closed bounds=[-9, 9]

signed slacks=[6, 12];inside inclusive interval=Yes

rectangle coordinate1 lower slack := sub(0, -8)

rectangle coordinate1 lower slack = 8

rectangle coordinate1 upper slack := sub(10, 0)

rectangle coordinate1 upper slack = 10

coordinate index=1;value=0;closed bounds=[-8, 10]

signed slacks=[8, 10];inside inclusive interval=Yes

Reverse checks

• 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 rectangle coordinate1 upper slack with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute rectangle coordinate1 lower slack 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 rectangle coordinate0 upper slack with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute rectangle coordinate0 lower slack with exact rational arithmetic, not rounded decimals.

Result

([-9, 9, -8, 10], [-3, 0], 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.