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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-20261005-advance-20261006-r01-order-relation-foundations · 2026-10-05 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Under the usual real order, compare (0)/1 and (0)/1. Both denominators are positive. Give <, =, or > in that order.
=
The exact cross products are 0 and 0, giving =. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-2)/3 and (-9)/2. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are -4 and -27, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (3)/4 and (-8)/3. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 9 and -32, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-4)/5 and (-5)/4. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are -16 and -25, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (5)/6 and (-6)/1. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 5 and -36, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-6)/2 and (-5)/2. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -12 and -10, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (7)/3 and (7)/3. Both denominators are positive. Give <, =, or > in that order.
=
The exact cross products are 21 and 21, giving =. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-8)/4 and (-3)/4. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -32 and -12, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (9)/5 and (-2)/1. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 9 and -10, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-10)/6 and (-11)/2. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are -20 and -66, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (11)/2 and (0)/3. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 33 and 0, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-12)/3 and (1)/4. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -48 and 3, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (13)/4 and (13)/4. Both denominators are positive. Give <, =, or > in that order.
=
The exact cross products are 52 and 52, giving =. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-14)/5 and (3)/2. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -28 and 15, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (15)/6 and (4)/3. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 45 and 24, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-16)/2 and (-17)/4. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -64 and -34, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (17)/3 and (6)/1. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are 17 and 18, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-18)/4 and (7)/2. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -36 and 28, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (19)/5 and (19)/5. Both denominators are positive. Give <, =, or > in that order.
=
The exact cross products are 95 and 95, giving =. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-20)/6 and (9)/4. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -80 and 54, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (21)/2 and (10)/1. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 21 and 20, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-22)/3 and (-23)/2. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are -44 and -69, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (23)/4 and (12)/3. Both denominators are positive. Give <, =, or > in that order.
>
The exact cross products are 69 and 48, giving >. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (-24)/5 and (13)/4. Both denominators are positive. Give <, =, or > in that order.
<
The exact cross products are -96 and 65, giving <. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, compare (25)/6 and (25)/6. Both denominators are positive. Give <, =, or > in that order.
=
The exact cross products are 150 and 150, giving =. Positive denominators preserve the comparison direction.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-rational-comparison · candidate
Under the usual real order, the explicit finite nonempty set has elements {0}. Give (minimum, maximum).
(0, 0)
Order the given elements to obtain (0, 0). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-1, 0, 3}. Give (minimum, maximum).
(-1, 3)
Order the given elements to obtain (-1, 3). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-2, 0, 4}. Give (minimum, maximum).
(-2, 4)
Order the given elements to obtain (-2, 4). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-3, 0, 5}. Give (minimum, maximum).
(-3, 5)
Order the given elements to obtain (-3, 5). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-7, -6, -4}. Give (minimum, maximum).
(-7, -4)
Order the given elements to obtain (-7, -4). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-5, 0, 7}. Give (minimum, maximum).
(-5, 7)
Order the given elements to obtain (-5, 7). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-6, 0, 8}. Give (minimum, maximum).
(-6, 8)
Order the given elements to obtain (-6, 8). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-7, 0, 9}. Give (minimum, maximum).
(-7, 9)
Order the given elements to obtain (-7, 9). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-11, -10, -8}. Give (minimum, maximum).
(-11, -8)
Order the given elements to obtain (-11, -8). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-9, 0, 11}. Give (minimum, maximum).
(-9, 11)
Order the given elements to obtain (-9, 11). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-10, 0, 12}. Give (minimum, maximum).
(-10, 12)
Order the given elements to obtain (-10, 12). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-11, 0, 13}. Give (minimum, maximum).
(-11, 13)
Order the given elements to obtain (-11, 13). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-15, -14, -12}. Give (minimum, maximum).
(-15, -12)
Order the given elements to obtain (-15, -12). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-13, 0, 15}. Give (minimum, maximum).
(-13, 15)
Order the given elements to obtain (-13, 15). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-14, 0, 16}. Give (minimum, maximum).
(-14, 16)
Order the given elements to obtain (-14, 16). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-15, 0, 17}. Give (minimum, maximum).
(-15, 17)
Order the given elements to obtain (-15, 17). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-19, -18, -16}. Give (minimum, maximum).
(-19, -16)
Order the given elements to obtain (-19, -16). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-17, 0, 19}. Give (minimum, maximum).
(-17, 19)
Order the given elements to obtain (-17, 19). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-18, 0, 20}. Give (minimum, maximum).
(-18, 20)
Order the given elements to obtain (-18, 20). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-19, 0, 21}. Give (minimum, maximum).
(-19, 21)
Order the given elements to obtain (-19, 21). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-23, -22, -20}. Give (minimum, maximum).
(-23, -20)
Order the given elements to obtain (-23, -20). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-21, 0, 23}. Give (minimum, maximum).
(-21, 23)
Order the given elements to obtain (-21, 23). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-22, 0, 24}. Give (minimum, maximum).
(-22, 24)
Order the given elements to obtain (-22, 24). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-23, 0, 25}. Give (minimum, maximum).
(-23, 25)
Order the given elements to obtain (-23, 25). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, the explicit finite nonempty set has elements {-27, -26, -24}. Give (minimum, maximum).
(-27, -24)
Order the given elements to obtain (-27, -24). Listing order is irrelevant; a singleton has equal extrema, while this rule does not assign extrema to an empty set.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-finite-extrema · candidate
Under the usual real order, is (0)/1 in the closed interval [0, 0]? The denominator is positive and both endpoints are included.
Yes
Check 0·1≤0≤0·1 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (6)/2 in the closed interval [-1, 3]? The denominator is positive and both endpoints are included.
Yes
Check -1·2≤6≤3·2 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-7)/3 in the closed interval [-2, 4]? The denominator is positive and both endpoints are included.
No
Check -2·3≤-7≤4·3 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (21)/4 in the closed interval [-3, 5]? The denominator is positive and both endpoints are included.
No
Check -3·4≤21≤5·4 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (1)/1 in the closed interval [-4, 6]? The denominator is positive and both endpoints are included.
Yes
Check -4·1≤1≤6·1 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-10)/2 in the closed interval [-5, 7]? The denominator is positive and both endpoints are included.
Yes
Check -5·2≤-10≤7·2 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (24)/3 in the closed interval [-6, 8]? The denominator is positive and both endpoints are included.
Yes
Check -6·3≤24≤8·3 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-29)/4 in the closed interval [-7, 9]? The denominator is positive and both endpoints are included.
No
Check -7·4≤-29≤9·4 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (11)/1 in the closed interval [-8, 10]? The denominator is positive and both endpoints are included.
No
Check -8·1≤11≤10·1 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (1)/2 in the closed interval [-9, 11]? The denominator is positive and both endpoints are included.
Yes
Check -9·2≤1≤11·2 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-30)/3 in the closed interval [-10, 12]? The denominator is positive and both endpoints are included.
Yes
Check -10·3≤-30≤12·3 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (52)/4 in the closed interval [-11, 13]? The denominator is positive and both endpoints are included.
Yes
Check -11·4≤52≤13·4 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-13)/1 in the closed interval [-12, 14]? The denominator is positive and both endpoints are included.
No
Check -12·1≤-13≤14·1 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (31)/2 in the closed interval [-13, 15]? The denominator is positive and both endpoints are included.
No
Check -13·2≤31≤15·2 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (1)/3 in the closed interval [-14, 16]? The denominator is positive and both endpoints are included.
Yes
Check -14·3≤1≤16·3 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-60)/4 in the closed interval [-15, 17]? The denominator is positive and both endpoints are included.
Yes
Check -15·4≤-60≤17·4 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (18)/1 in the closed interval [-16, 18]? The denominator is positive and both endpoints are included.
Yes
Check -16·1≤18≤18·1 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-35)/2 in the closed interval [-17, 19]? The denominator is positive and both endpoints are included.
No
Check -17·2≤-35≤19·2 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (61)/3 in the closed interval [-18, 20]? The denominator is positive and both endpoints are included.
No
Check -18·3≤61≤20·3 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (1)/4 in the closed interval [-19, 21]? The denominator is positive and both endpoints are included.
Yes
Check -19·4≤1≤21·4 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-20)/1 in the closed interval [-20, 22]? The denominator is positive and both endpoints are included.
Yes
Check -20·1≤-20≤22·1 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (46)/2 in the closed interval [-21, 23]? The denominator is positive and both endpoints are included.
Yes
Check -21·2≤46≤23·2 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (-67)/3 in the closed interval [-22, 24]? The denominator is positive and both endpoints are included.
No
Check -22·3≤-67≤24·3 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (101)/4 in the closed interval [-23, 25]? The denominator is positive and both endpoints are included.
No
Check -23·4≤101≤25·4 exactly, obtaining No. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, is (1)/1 in the closed interval [-24, 26]? The denominator is positive and both endpoints are included.
Yes
Check -24·1≤1≤26·1 exactly, obtaining Yes. A singleton closed interval still contains its endpoint.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-closed-interval · candidate
Under the usual real order, a=0, b=0, and c=-12. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -12 and -12, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-3, b=2, and c=-11. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and -9, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-4, b=-8, and c=-10. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -18, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-5, b=-5, and c=-9. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-6, b=5, and c=-8. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and -3, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-7, b=-14, and c=-7. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -21, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-8, b=-8, and c=-6. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-9, b=8, and c=-5. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and 3, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-10, b=-20, and c=-4. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -24, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-11, b=-11, and c=-3. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-12, b=11, and c=-2. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and 9, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-13, b=-26, and c=-1. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -27, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-14, b=-14, and c=0. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-15, b=14, and c=1. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and 15, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-16, b=-32, and c=2. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -30, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-17, b=-17, and c=3. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-18, b=17, and c=4. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and 21, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-19, b=-38, and c=5. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -33, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-20, b=-20, and c=6. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-21, b=20, and c=7. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and 27, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-22, b=-44, and c=8. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -36, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-23, b=-23, and c=9. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-24, b=23, and c=10. Compare a+c and b+c, giving <, =, or > in that order.
<
The translated values are -14 and 33, with relation <. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-25, b=-50, and c=11. Compare a+c and b+c, giving <, =, or > in that order.
>
The translated values are -14 and -39, with relation >. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
Under the usual real order, a=-26, b=-26, and c=12. Compare a+c and b+c, giving <, =, or > in that order.
=
The translated values are -14 and -14, with relation =. A common addend preserves the original order, including equality.
Reviewed step-by-step derivation
Usual order on the real numbers · qf-order-common-translation · candidate
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