Environments, Simultaneous Substitution, and Original Domains
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261008-advance-20261011-r03-variable-expression-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(0)x+(-3) is an expression, not an equation. Ordered x environments=[0/3, -1/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-3/1, -3/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-3/1, -3/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
2. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-1)x+(-6) is an expression, not an equation. Ordered x environments=[1/3, -2/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-19/3, -5/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-19/3, -5/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
3. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(2)x+(-5) is an expression, not an equation. Ordered x environments=[2/3, -3/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-11/3, -8/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-11/3, -8/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
4. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-3)x+(-4) is an expression, not an equation. Ordered x environments=[3/3, -4/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-7/1, 2/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-7/1, 2/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
5. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(4)x+(-3) is an expression, not an equation. Ordered x environments=[4/3, -5/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(7/3, -13/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (7/3, -13/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
6. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-5)x+(-2) is an expression, not an equation. Ordered x environments=[5/3, -6/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-31/3, 13/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-31/3, 13/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
7. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(6)x+(-1) is an expression, not an equation. Ordered x environments=[6/3, -7/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(11/1, -22/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (11/1, -22/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
8. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-7)x+(0) is an expression, not an equation. Ordered x environments=[7/3, -8/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-49/3, 28/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-49/3, 28/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
9. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(8)x+(1) is an expression, not an equation. Ordered x environments=[8/3, -9/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(67/3, -35/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (67/3, -35/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
10. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-9)x+(2) is an expression, not an equation. Ordered x environments=[9/3, -10/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-25/1, 47/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-25/1, 47/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
11. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(10)x+(3) is an expression, not an equation. Ordered x environments=[10/3, -11/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(109/3, -52/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (109/3, -52/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
12. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-11)x+(4) is an expression, not an equation. Ordered x environments=[11/3, -12/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-109/3, 70/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-109/3, 70/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
13. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(12)x+(5) is an expression, not an equation. Ordered x environments=[12/3, -13/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(53/1, -73/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (53/1, -73/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
14. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-13)x+(6) is an expression, not an equation. Ordered x environments=[13/3, -14/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-151/3, 97/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-151/3, 97/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
15. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(14)x+(7) is an expression, not an equation. Ordered x environments=[14/3, -15/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(217/3, -98/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (217/3, -98/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
16. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-15)x+(8) is an expression, not an equation. Ordered x environments=[15/3, -16/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-67/1, 128/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-67/1, 128/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
17. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(16)x+(9) is an expression, not an equation. Ordered x environments=[16/3, -17/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(283/3, -127/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (283/3, -127/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
18. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-17)x+(10) is an expression, not an equation. Ordered x environments=[17/3, -18/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-259/3, 163/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-259/3, 163/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
19. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(18)x+(11) is an expression, not an equation. Ordered x environments=[18/3, -19/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(119/1, -160/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (119/1, -160/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
20. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-19)x+(12) is an expression, not an equation. Ordered x environments=[19/3, -20/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-325/3, 202/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-325/3, 202/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
21. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(20)x+(13) is an expression, not an equation. Ordered x environments=[20/3, -21/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(439/3, -197/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (439/3, -197/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
22. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-21)x+(14) is an expression, not an equation. Ordered x environments=[21/3, -22/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-133/1, 245/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-133/1, 245/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
23. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(22)x+(15) is an expression, not an equation. Ordered x environments=[22/3, -23/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(529/3, -238/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (529/3, -238/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
24. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-23)x+(16) is an expression, not an equation. Ordered x environments=[23/3, -24/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-481/3, 292/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-481/3, 292/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
25. Affine Expression in Two Environments
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(24)x+(17) is an expression, not an equation. Ordered x environments=[24/3, -25/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(209/1, -283/1)
Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (209/1, -283/1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate
26. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(0)x+(0)y, x=1, y=2. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(0, 0)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
27. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(2)x+(2)y, x=-1, y=3. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(4, 4)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (4, 4).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
28. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(1)x+(-2)y, x=3, y=3. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-3, -3)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-3, -3).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
29. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(4)x+(-3)y, x=1, y=6. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-14, 21)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-14, 21).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
30. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(5)x+(-4)y, x=2, y=7. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-18, 27)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-18, 27).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
31. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(6)x+(-5)y, x=-2, y=8. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-52, 58)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-52, 58).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
32. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(7)x+(-6)y, x=-1, y=9. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-61, 69)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-61, 69).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
33. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(8)x+(-7)y, x=0, y=10. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-70, 80)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-70, 80).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
34. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(9)x+(-8)y, x=1, y=11. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-79, 91)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-79, 91).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
35. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(10)x+(-9)y, x=2, y=12. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-88, 102)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-88, 102).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
36. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(11)x+(-10)y, x=-2, y=13. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-152, 163)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-152, 163).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
37. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(12)x+(-11)y, x=-1, y=14. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-166, 179)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-166, 179).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
38. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(13)x+(-12)y, x=0, y=15. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-180, 195)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-180, 195).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
39. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(14)x+(-13)y, x=1, y=16. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-194, 211)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-194, 211).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
40. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(15)x+(-14)y, x=2, y=17. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-208, 227)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-208, 227).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
41. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(16)x+(-15)y, x=-2, y=18. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-302, 318)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-302, 318).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
42. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(17)x+(-16)y, x=-1, y=19. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-321, 339)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-321, 339).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
43. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(18)x+(-17)y, x=0, y=20. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-340, 360)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-340, 360).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
44. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(19)x+(-18)y, x=1, y=21. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-359, 381)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-359, 381).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
45. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(20)x+(-19)y, x=2, y=22. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-378, 402)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-378, 402).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
46. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(21)x+(-20)y, x=-2, y=23. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-502, 523)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-502, 523).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
47. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(22)x+(-21)y, x=-1, y=24. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-526, 549)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-526, 549).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
48. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(23)x+(-22)y, x=0, y=25. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-550, 575)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-550, 575).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
49. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(24)x+(-23)y, x=1, y=26. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-574, 601)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-574, 601).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
50. Simultaneous Variable Swap
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(25)x+(-24)y, x=2, y=27. Swap x and y simultaneously; give (original value, swapped value) as integers.
View answer and explanation
(-598, 627)
Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-598, 627).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate
51. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=0, locally shadowing inner x=0, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(1, 1)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
52. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=3, locally shadowing inner x=-2, c=4. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(7, 2)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (7, 2).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
53. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-10, locally shadowing inner x=-1, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-10, -1)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-10, -1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
54. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-9, locally shadowing inner x=-2, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-8, -1)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-8, -1).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
55. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-8, locally shadowing inner x=-3, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-10, -5)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-10, -5).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
56. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-7, locally shadowing inner x=-4, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-8, -5)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-8, -5).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
57. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-6, locally shadowing inner x=-5, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-6, -5)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-6, -5).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
58. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-5, locally shadowing inner x=-6, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-4, -5)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-4, -5).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
59. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-4, locally shadowing inner x=-7, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-6, -9)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-6, -9).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
60. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-3, locally shadowing inner x=-8, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-4, -9)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-4, -9).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
61. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-2, locally shadowing inner x=-9, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-2, -9)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-2, -9).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
62. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-1, locally shadowing inner x=-10, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(0, -9)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (0, -9).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
63. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=0, locally shadowing inner x=-11, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(-2, -13)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-2, -13).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
64. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=1, locally shadowing inner x=-12, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(0, -13)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (0, -13).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
65. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=2, locally shadowing inner x=-13, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(2, -13)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (2, -13).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
66. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=3, locally shadowing inner x=-14, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(4, -13)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (4, -13).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
67. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=4, locally shadowing inner x=-15, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(2, -17)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (2, -17).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
68. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=5, locally shadowing inner x=-16, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(4, -17)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (4, -17).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
69. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=6, locally shadowing inner x=-17, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(6, -17)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (6, -17).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
70. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=7, locally shadowing inner x=-18, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(8, -17)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (8, -17).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
71. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=8, locally shadowing inner x=-19, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(6, -21)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (6, -21).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
72. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=9, locally shadowing inner x=-20, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(8, -21)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (8, -21).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
73. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=10, locally shadowing inner x=-21, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(10, -21)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (10, -21).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
74. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=11, locally shadowing inner x=-22, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(12, -21)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (12, -21).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
75. Outer and Lexically Shadowed Values
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=12, locally shadowing inner x=-23, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.
View answer and explanation
(10, -25)
Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (10, -25).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate
76. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(0)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 2/1, 0/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 0/1, 0/1])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 0/1, 0/1]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
77. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(5)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, -2/1, -2/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, No, No], [undefined, undefined, undefined])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, No, No], [undefined, undefined, undefined]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
78. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(2)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 1/1, -1/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 2/1, -2/1])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 2/1, -2/1]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
79. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(2)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 6/2, -4/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 1/1, -2/5])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 1/1, -2/5]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
80. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(3)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 9/2, -4/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 6/5, -1/2])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 6/5, -1/2]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
81. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(4)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 2/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 4/3, -4/7])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 4/3, -4/7]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
82. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(5)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 5/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 10/7, -5/8])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 10/7, -5/8]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
83. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(6)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 8/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 3/2, -2/3])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 3/2, -2/3]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
84. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(7)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 11/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 14/9, -7/10])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 14/9, -7/10]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
85. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(8)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 14/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 8/5, -8/11])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 8/5, -8/11]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
86. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(9)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 7/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 18/11, -3/4])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 18/11, -3/4]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
87. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(10)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 10/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 5/3, -10/13])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 5/3, -10/13]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
88. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(11)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 13/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 22/13, -11/14])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 22/13, -11/14]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
89. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(12)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 16/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 12/7, -4/5])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 12/7, -4/5]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
90. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(13)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 19/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 26/15, -13/16])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 26/15, -13/16]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
91. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(14)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 12/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 7/4, -14/17])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 7/4, -14/17]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
92. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(15)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 15/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 30/17, -5/6])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 30/17, -5/6]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
93. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(16)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 18/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 16/9, -16/19])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 16/9, -16/19]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
94. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(17)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 21/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 34/19, -17/20])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 34/19, -17/20]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
95. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(18)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 24/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 9/5, -6/7])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 9/5, -6/7]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
96. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(19)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 17/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 38/21, -19/22])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 38/21, -19/22]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
97. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(20)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 20/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 20/11, -20/23])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 20/11, -20/23]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
98. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(21)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 23/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 42/23, -7/8])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 42/23, -7/8]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
99. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(22)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 26/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 11/6, -22/25])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 11/6, -22/25]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
100. Original Rational-Expression Domain Guard
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(23)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 29/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).
View answer and explanation
([No, Yes, Yes], [undefined, 46/25, -23/26])
Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 46/25, -23/26]).
Reviewed step-by-step derivation
Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate
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