# Exact Rational Midpoints and Closed Intervals

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261010-advance-20261014-r02-real-number-foundations · 2026-10-10 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=0/1≤b=0/1. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(0/1, 0/1, 0/1)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 0/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0001/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 2. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-1/2≤b=1/2. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(0/1, 1/2, 1/2)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 1/2, 1/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0002/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 3. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-2/2≤b=3/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(0/1, 1/1, 1/1)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 1/1, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0003/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 4. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-3/2≤b=4/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/12, 17/12, 17/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/12, 17/12, 17/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0004/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 5. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-4/2≤b=5/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/6, 11/6, 11/6)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/6, 11/6, 11/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0005/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 6. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-5/2≤b=6/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/4, 9/4, 9/4)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/4, 9/4, 9/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0006/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 7. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-6/2≤b=7/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/3, 8/3, 8/3)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/3, 8/3, 8/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0007/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 8. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-7/2≤b=8/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/12, 37/12, 37/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/12, 37/12, 37/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0008/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 9. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-8/2≤b=9/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/2, 7/2, 7/2)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/2, 7/2, 7/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0009/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 10. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-9/2≤b=10/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7/12, 47/12, 47/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/12, 47/12, 47/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0010/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 11. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-10/2≤b=11/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2/3, 13/3, 13/3)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-2/3, 13/3, 13/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0011/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 12. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-11/2≤b=12/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/4, 19/4, 19/4)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-3/4, 19/4, 19/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0012/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 13. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-12/2≤b=13/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/6, 31/6, 31/6)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/6, 31/6, 31/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0013/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 14. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-13/2≤b=14/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-11/12, 67/12, 67/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-11/12, 67/12, 67/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0014/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 15. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-14/2≤b=15/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1/1, 6/1, 6/1)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/1, 6/1, 6/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0015/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 16. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-15/2≤b=16/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-13/12, 77/12, 77/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-13/12, 77/12, 77/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0016/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 17. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-16/2≤b=17/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7/6, 41/6, 41/6)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/6, 41/6, 41/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0017/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 18. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-17/2≤b=18/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/4, 29/4, 29/4)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/4, 29/4, 29/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0018/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 19. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-18/2≤b=19/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4/3, 23/3, 23/3)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-4/3, 23/3, 23/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0019/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 20. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-19/2≤b=20/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-17/12, 97/12, 97/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-17/12, 97/12, 97/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0020/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 21. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-20/2≤b=21/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/2, 17/2, 17/2)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-3/2, 17/2, 17/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0021/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 22. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-21/2≤b=22/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-19/12, 107/12, 107/12)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-19/12, 107/12, 107/12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0022/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 23. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-22/2≤b=23/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-5/3, 28/3, 28/3)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/3, 28/3, 28/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0023/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 24. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-23/2≤b=24/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7/4, 39/4, 39/4)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/4, 39/4, 39/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0024/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 25. Ordered Rational Midpoint and Endpoint Gaps

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-24/2≤b=25/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-11/6, 61/6, 61/6)

Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-11/6, 61/6, 61/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0025/)

Real number · qf-density14-real-ordered-midpoint · candidate





## 26. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[0/1, 0/-2, 0/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([0/1], 0/1, 0/1, 0/1)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([0/1], 0/1, 0/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0026/)

Real number · qf-density14-real-point-closed-hull · candidate





## 27. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[1/2, -1/2, 1/2]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-1/2, 1/2], -1/2, 1/2, 1/1)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-1/2, 1/2], -1/2, 1/2, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0027/)

Real number · qf-density14-real-point-closed-hull · candidate





## 28. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-2/2, 0/1, 3/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0028/)

Real number · qf-density14-real-point-closed-hull · candidate





## 29. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-3/2, 0/1, 4/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0029/)

Real number · qf-density14-real-point-closed-hull · candidate





## 30. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-4/2, 0/1, 5/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0030/)

Real number · qf-density14-real-point-closed-hull · candidate





## 31. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-5/2, 0/1, 6/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0031/)

Real number · qf-density14-real-point-closed-hull · candidate





## 32. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-6/2, 0/1, 7/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0032/)

Real number · qf-density14-real-point-closed-hull · candidate





## 33. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-7/2, 0/1, 8/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0033/)

Real number · qf-density14-real-point-closed-hull · candidate





## 34. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-8/2, 0/1, 9/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0034/)

Real number · qf-density14-real-point-closed-hull · candidate





## 35. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-9/2, 0/1, 10/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0035/)

Real number · qf-density14-real-point-closed-hull · candidate





## 36. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-10/2, 0/1, 11/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0036/)

Real number · qf-density14-real-point-closed-hull · candidate





## 37. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-11/2, 0/1, 12/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0037/)

Real number · qf-density14-real-point-closed-hull · candidate





## 38. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-12/2, 0/1, 13/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0038/)

Real number · qf-density14-real-point-closed-hull · candidate





## 39. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-13/2, 0/1, 14/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0039/)

Real number · qf-density14-real-point-closed-hull · candidate





## 40. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-14/2, 0/1, 15/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0040/)

Real number · qf-density14-real-point-closed-hull · candidate





## 41. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-15/2, 0/1, 16/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0041/)

Real number · qf-density14-real-point-closed-hull · candidate





## 42. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-16/2, 0/1, 17/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0042/)

Real number · qf-density14-real-point-closed-hull · candidate





## 43. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-17/2, 0/1, 18/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0043/)

Real number · qf-density14-real-point-closed-hull · candidate





## 44. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-18/2, 0/1, 19/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0044/)

Real number · qf-density14-real-point-closed-hull · candidate





## 45. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-19/2, 0/1, 20/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0045/)

Real number · qf-density14-real-point-closed-hull · candidate





## 46. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-20/2, 0/1, 21/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0046/)

Real number · qf-density14-real-point-closed-hull · candidate





## 47. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-21/2, 0/1, 22/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0047/)

Real number · qf-density14-real-point-closed-hull · candidate





## 48. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-22/2, 0/1, 23/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0048/)

Real number · qf-density14-real-point-closed-hull · candidate





## 49. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-23/2, 0/1, 24/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0049/)

Real number · qf-density14-real-point-closed-hull · candidate





## 50. Finite Distinct Points and Least Containing Closed Interval

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-24/2, 0/1, 25/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.

View answer and explanation([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3)

Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0050/)

Real number · qf-density14-real-point-closed-hull · candidate





## 51. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[1/1, 2/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [1/1, 1/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [1/1, 1/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0051/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 52. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[2/1, 3/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(No, [], 1/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (No, [], 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0052/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 53. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 0/1], J=[0/1, 1/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 0/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 0/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0053/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 54. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-3/2, 3/2], J=[0/1, 5/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 3/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 3/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0054/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 55. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-4/2, 4/2], J=[0/1, 6/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 2/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 2/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0055/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 56. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-5/2, 5/2], J=[0/1, 7/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 5/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 5/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0056/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 57. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-6/2, 6/2], J=[0/1, 8/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 3/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 3/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0057/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 58. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-7/2, 7/2], J=[0/1, 9/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 7/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 7/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0058/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 59. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-8/2, 8/2], J=[0/1, 10/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 4/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 4/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0059/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 60. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-9/2, 9/2], J=[0/1, 11/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 9/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 9/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0060/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 61. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-10/2, 10/2], J=[0/1, 12/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 5/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 5/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0061/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 62. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-11/2, 11/2], J=[0/1, 13/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 11/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 11/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0062/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 63. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-12/2, 12/2], J=[0/1, 14/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 6/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 6/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0063/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 64. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-13/2, 13/2], J=[0/1, 15/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 13/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 13/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0064/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 65. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-14/2, 14/2], J=[0/1, 16/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 7/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 7/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0065/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 66. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-15/2, 15/2], J=[0/1, 17/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 15/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 15/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0066/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 67. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-16/2, 16/2], J=[0/1, 18/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 8/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 8/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0067/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 68. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-17/2, 17/2], J=[0/1, 19/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 17/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 17/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0068/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 69. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-18/2, 18/2], J=[0/1, 20/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 9/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 9/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0069/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 70. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-19/2, 19/2], J=[0/1, 21/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 19/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 19/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0070/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 71. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-20/2, 20/2], J=[0/1, 22/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 10/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 10/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0071/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 72. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-21/2, 21/2], J=[0/1, 23/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 21/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 21/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0072/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 73. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-22/2, 22/2], J=[0/1, 24/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 11/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 11/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0073/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 74. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-23/2, 23/2], J=[0/1, 25/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 23/2], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 23/2], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0074/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 75. Closed Interval Intersection and Separation Gap

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-24/2, 24/2], J=[0/1, 26/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(Yes, [0/1, 12/1], 0/1)

Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 12/1], 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0075/)

Real number · qf-density14-real-overlap-closed-intervals · candidate





## 76. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-12/2, ordered points=[-10/1, -13/2, -6/1, -16/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0076/)

Real number · qf-density14-real-point-offset-distances · candidate





## 77. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-11/2, ordered points=[-55/6, -6/1, -11/2, -29/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0077/)

Real number · qf-density14-real-point-offset-distances · candidate





## 78. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-10/2, ordered points=[-25/3, -11/2, -5/1, -13/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0078/)

Real number · qf-density14-real-point-offset-distances · candidate





## 79. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-9/2, ordered points=[-15/2, -5/1, -9/2, -23/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0079/)

Real number · qf-density14-real-point-offset-distances · candidate





## 80. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-8/2, ordered points=[-20/3, -9/2, -4/1, -10/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0080/)

Real number · qf-density14-real-point-offset-distances · candidate





## 81. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-7/2, ordered points=[-35/6, -4/1, -7/2, -17/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0081/)

Real number · qf-density14-real-point-offset-distances · candidate





## 82. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-6/2, ordered points=[-5/1, -7/2, -3/1, -7/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0082/)

Real number · qf-density14-real-point-offset-distances · candidate





## 83. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-5/2, ordered points=[-25/6, -3/1, -5/2, -11/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0083/)

Real number · qf-density14-real-point-offset-distances · candidate





## 84. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-4/2, ordered points=[-10/3, -5/2, -2/1, -4/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0084/)

Real number · qf-density14-real-point-offset-distances · candidate





## 85. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-3/2, ordered points=[-5/2, -2/1, -3/2, -5/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0085/)

Real number · qf-density14-real-point-offset-distances · candidate





## 86. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-2/2, ordered points=[-5/3, -3/2, -1/1, -1/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0086/)

Real number · qf-density14-real-point-offset-distances · candidate





## 87. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-1/2, ordered points=[-5/6, -1/1, -1/2, 1/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0087/)

Real number · qf-density14-real-point-offset-distances · candidate





## 88. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=0/2, ordered points=[0/1, -1/2, 0/1, 2/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0088/)

Real number · qf-density14-real-point-offset-distances · candidate





## 89. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=1/2, ordered points=[5/6, 0/1, 1/2, 7/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0089/)

Real number · qf-density14-real-point-offset-distances · candidate





## 90. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=2/2, ordered points=[5/3, 1/2, 1/1, 5/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0090/)

Real number · qf-density14-real-point-offset-distances · candidate





## 91. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=3/2, ordered points=[5/2, 1/1, 3/2, 13/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0091/)

Real number · qf-density14-real-point-offset-distances · candidate





## 92. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=4/2, ordered points=[10/3, 3/2, 2/1, 8/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0092/)

Real number · qf-density14-real-point-offset-distances · candidate





## 93. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=5/2, ordered points=[25/6, 2/1, 5/2, 19/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0093/)

Real number · qf-density14-real-point-offset-distances · candidate





## 94. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=6/2, ordered points=[5/1, 5/2, 3/1, 11/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0094/)

Real number · qf-density14-real-point-offset-distances · candidate





## 95. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=7/2, ordered points=[35/6, 3/1, 7/2, 25/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0095/)

Real number · qf-density14-real-point-offset-distances · candidate





## 96. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=8/2, ordered points=[20/3, 7/2, 4/1, 14/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0096/)

Real number · qf-density14-real-point-offset-distances · candidate





## 97. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=9/2, ordered points=[15/2, 4/1, 9/2, 31/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0097/)

Real number · qf-density14-real-point-offset-distances · candidate





## 98. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=10/2, ordered points=[25/3, 9/2, 5/1, 17/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0098/)

Real number · qf-density14-real-point-offset-distances · candidate





## 99. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=11/2, ordered points=[55/6, 5/1, 11/2, 37/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0099/)

Real number · qf-density14-real-point-offset-distances · candidate





## 100. Ordered Rational Offsets and Absolute Distances

Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=12/2, ordered points=[10/1, 11/2, 6/1, 20/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.

View answer and explanation([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])

Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r02-real-number-0100/)

Real number · qf-density14-real-point-offset-distances · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r02-real-number-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261010-advance-20261014-r02-real-number-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261010-advance-20261014-r02-real-number-foundations/validation-report.json)