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If -2 < a < 11 and 3 < b < 12, then which of the following is NOT always true?
A. 1 < a + b < 23
B. -14 < a - b < 8
C. -7 < b - a < 14
D. 1 < b + a < 23
E. -24 < a b < 132
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A. 1 < a + b < 23
B. -14 < a - b < 8
C. -7 < b - a < 14
D. 1 < b + a < 23
E. -24 < a b < 132
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Bunuel
Hi....
We will be playing around with Max and min values of both a and b depending on the operation we do..
Although a+B is same as B+a, and choices A and D are same, we need not see these...
a+b...
For range, take max values of both for upper end and min values for lower end.
upper end : 11+12=23 & lower end : -2+3=1.... So 1<a+b<11
A & D are correct
a-b...
For range, take max value of a and min value of b for upper end and min value of a and max value of b for lower end.
upper end : 11-3=8 & lower end : -2-12=-14.... So -14<a-b<8
B is correct
b-a.
For range, take min value of a and max value of b for upper end and max value of a and min value of b for lower end.
upper end : 12-(-2)=14 & lower end : 3-11=-8.... So -8<b-a<14 but given is -7<b-a<14
C is Not correct
ab...
For range, take max value of a and max value of b for upper end and min value of a and max value of b for lower end.
upper end : 11*12=132 & lower end : -2*12=-24... So -24<ab<132
E is correct ..
C is the answer
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General Discussion
Posts: 1,142
We have range of a: -2 < a < 11. This means range of -a: -11 < -a < 2
We have range of b: 3 < b < 12. This means range of -b: -12 < -b < -3
Now, range of (a+b) or (b+a): (-2+3) < a+b < (11+12) Or 1 < a+b < 23.
So A and D are true.
Range of a-b is range of a+(-b): (-2-12) < a-b < (11-3) Or -14 < a-b < 8
So B is also true
Range of b-a is range of b+(-a): (3-11) < b-a < (12+2) Or -8 < b-a < 14
As we can see, C is not always true
Hence C answer
We have range of b: 3 < b < 12. This means range of -b: -12 < -b < -3
Now, range of (a+b) or (b+a): (-2+3) < a+b < (11+12) Or 1 < a+b < 23.
So A and D are true.
Range of a-b is range of a+(-b): (-2-12) < a-b < (11-3) Or -14 < a-b < 8
So B is also true
Range of b-a is range of b+(-a): (3-11) < b-a < (12+2) Or -8 < b-a < 14
As we can see, C is not always true
Hence C answer
Bunuel
A. 1 < a + b < 23 Using the values given we see that (a+b) ranges from (-2+3=1) to (11+12=23) hence This option is True
B. -14 < a - b < 8 Using the values given we see that (a-b) ranges from (-2-12=-14) to (11-3=8) hence This option is True
C. -7 < b - a < 14 Using the values given we see that (b-a) ranges from (3-11=-8) to (12-(-2)=14) hence This option is NOT true for lowest value of (b-a)
D. 1 < b + a < 23 Using the values given we see that (a+b) ranges from (-2+3=1) to (11+12=23) hence This option is True
E. -24 < a b < 132 Using the values given we see that (a.b) ranges from (-2*12=-24) to (11*12=132) hence This option is True
ANswer: option C
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selim
Ans: C
My method is a bit calculative we need to find which one is not true and for that we need to do the calculation
for A: a's min + b's min <a+b< a's max + b's max = -2+3 <a+b<11+12= 1<a+b<23 True
for B: a's min - b'max <a-b< a's max- b's min= -2-12 <a-b< 11-3 = -14<a-b<8 True
for C: b's min-a' max <b-a<b's max- a's min= 3-11 <b-a<12-(-2) = -8<b-a<14 Not True
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Expert reply
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selim
If we add the two inequalities together, we have:
1 < a + b < 23
Thus, A is true and since a + b = b + a, D is also true.
Multiplying the second inequality by -1, we have 3 > -b > -12 or -12 < -b < 3. Now, adding the latter to the first inequality, we have:
–14 < a – b < 8
So B is true.
Similarly, multiplying the first inequality by -1, we have 2 > -a > -11 or -11 < -a < 2. Now. adding the latter to the second inequality, we have:
–8 < b – a < 14
So C is NOT true since (b - a) could be -7.5, which does not fall into -7 < b - a < 14.
Answer: C
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Pankaj0901
Joined: 18 Dec 2018
Last visit: 17 Dec 2022
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Location: India
WE:Account Management (Hospitality and Tourism)
Posts: 399
I believe the wording of question is not correct, and can definitely be better to remove ambiguity.
If the range of b-a is: -8 to 14 (say, SET A) ...as explained in the above comments
Option C: -7 to 14 (say, SET B)
=> SET B is a subset of SET A. Hence, for all the values in range -7 to 14 (Option C), the value of b-a (range: -8 to 14) holds correctly.
Please note: Question asks to select an option which is NOT always true. But the option C will always hold true as it is the subset of A. Hence, Option C can surely not be the answer to this question.
Request Admin to kindly check this and correct the question wording.
If the range of b-a is: -8 to 14 (say, SET A) ...as explained in the above comments
Option C: -7 to 14 (say, SET B)
=> SET B is a subset of SET A. Hence, for all the values in range -7 to 14 (Option C), the value of b-a (range: -8 to 14) holds correctly.
Please note: Question asks to select an option which is NOT always true. But the option C will always hold true as it is the subset of A. Hence, Option C can surely not be the answer to this question.
Request Admin to kindly check this and correct the question wording.
PANKAJ0901
This is not true. If b = 3.1 and a = 10.9, then b - a = 3.1 - 10.9 = -7.8.
P.S. Please do not use report button. A post is enough.
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PANKAJ0901
Taking your observation further.
Rather, Set A has to be subset of Set B for the option to be true and not vice versa.
We know from Set A that b-a can also be something from -8 to -7, but Set B misses on it.
For example.
Say c is a prime number.
A) c is an integer would be true.
B) But c is odd prime number will not although it is a subset of prime number. If c=2, the option will not stand.
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chetan2u
Please find my remarks below in red and blue.
Taking your observation further.
Rather, Set A has to be subset of Set B for the option to be true and not vice versa.
We know from Set A that b-a can also be something from -8 to -7, but Set B misses on it.
For example.
Say c is a prime number. c={2,3,5,7,11....} Which of the following is NOT always true?
A) c is an integer would be true. {-3,-2,-1,0,1,...} are also integers, which need NOT ALWAYS be Prime numbers.
B) But c is odd prime number will not although it is a subset of prime number. {3,5,7,11,....} => Definitely true..as all these values are Prime numbers.
If c=2, the option will not stand. (That's Ok, but does c=2 make option B "WRONG" or "NOT RIGHT". I FAIL TO UNDERSTAND THIS!)
C) c is a complex number. => Not ALWAYS TRUE...similar to option A (as otherwise every solution in mathematics would be some complex number with imaginary part as 0 at most. Why should we even be zeroing down to a smaller set of range?)
D) c is a Natural number. => Not ALWAYS TRUE .....and so on....
I think, there are two ways to look these questions- and none of the two perspectives is completely wrong.
Thanks Bunuel and Chetan2u for the explanation (More Kudos to you guys). I understand that Option C doesn't capture the complete range, and that's not the point I am concerned with. It's clear that there could be more values of a,b (say, -10.9, 3.1, as mentioned by Bunuel). What I was trying to say here is something else. May be I need some break as despite a lot of thought I still couldn't quite comprehend.
As per the question, it says "..which of the following is NOT always true?" Option C is definitely always true! There could be more eligible values which are not captured in Option C (that's completely ok), my point is Option C is NOT WRONG. I had only pointed out for a clearer wording that doesn't leave any scope of ambiguity.
PS (For Bunuel)- Duly noted, report button will not be used. Thanks.
PANKAJ0901
The question is correct and there is only one correct interpretation
The question asks: If -2 < a < 11 and 3 < b < 12, then which of the following is NOT always true?
Only option C is NOT always true. I don't see any ambiguity whatsoever.
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Posts: 1,324
Given:
-2 < a < 11
and
3 < b < 12
The question is asking the following:
Which of the Answer Choice Ranges does NOT Always have to be true for EVERY Value that the expression can take?
-C-
-7 < b - a < 14
(1st) Find the MINIMUM Lower Bound of the expression (b - a) given the Ranges in the Q Prompt
We want to first Minimize the Value that (b - a) can take.
Since a is being subtracted from b ——-> we accomplish this by making b as LOW a VALUE as we can ——- and by making a as HIGH a VALUE as we can
The Lower Value Limit for b is: +3
The Upper Value Limit for a is: +11
Thus, the expression (b - a) can take Values greater than >
+3 - +11 = (-)8
(-)8 < (b - a)
However, the Range given by Answer C is:
(-)7 < (b - a) < +14
Answer C is NOT Always a True Statement.
For example, (b - a) can equal a Value like (-)7.5 or (-)7.6 and the Range in Answer Choice C would NOT cover these Values
Therefore, the answer is C
C does NOT always have to be True
Posted from my mobile device
-2 < a < 11
and
3 < b < 12
The question is asking the following:
Which of the Answer Choice Ranges does NOT Always have to be true for EVERY Value that the expression can take?
-C-
-7 < b - a < 14
(1st) Find the MINIMUM Lower Bound of the expression (b - a) given the Ranges in the Q Prompt
We want to first Minimize the Value that (b - a) can take.
Since a is being subtracted from b ——-> we accomplish this by making b as LOW a VALUE as we can ——- and by making a as HIGH a VALUE as we can
The Lower Value Limit for b is: +3
The Upper Value Limit for a is: +11
Thus, the expression (b - a) can take Values greater than >
+3 - +11 = (-)8
(-)8 < (b - a)
However, the Range given by Answer C is:
(-)7 < (b - a) < +14
Answer C is NOT Always a True Statement.
For example, (b - a) can equal a Value like (-)7.5 or (-)7.6 and the Range in Answer Choice C would NOT cover these Values
Therefore, the answer is C
C does NOT always have to be True
Posted from my mobile device
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Bunuel,
Could you please check what is wrong with my reasoning:
B) -14<a-b<8
if we take min of a-b, we get a little more than -14(a is more than -2, and b is less than 12)
if we take max of a-b, we get a little less than 8 (a is a little less than 11 and b is a little more than 3)
----> So B is always correct
E) -24<ab<132
If we take min of ab, we get a little more than -24 (a is a little larger than -2, b is a little smaller than 12)
If we take max of ab, we get a little less than 132 (a is a little smaller than 11, b is a little smaller than 12)
---> So E is also correct as per my logic. Could you please be so kind to point out my mistake. Much appreciated in advance!
Could you please check what is wrong with my reasoning:
B) -14<a-b<8
if we take min of a-b, we get a little more than -14(a is more than -2, and b is less than 12)
if we take max of a-b, we get a little less than 8 (a is a little less than 11 and b is a little more than 3)
----> So B is always correct
E) -24<ab<132
If we take min of ab, we get a little more than -24 (a is a little larger than -2, b is a little smaller than 12)
If we take max of ab, we get a little less than 132 (a is a little smaller than 11, b is a little smaller than 12)
---> So E is also correct as per my logic. Could you please be so kind to point out my mistake. Much appreciated in advance!
mira93
The question asks: ...which of the following is NOT always true?
A, B, D and E are always true.
E is not always true.
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Bunuel
A. a+b= 1 < a + b < 23
B. a-b = -14 < a - b < 8 [1,-5,8,-14 we take 8 and -14]
C. b-a= -8 < b - a < 14 [Given is -7 < b - a < 14 ] [1,5,-8,14 we can take 14 and -8]
D. 1 < b + a < 23 [Same as A]
E. -24 < a b < 132 [The four values are -6,132,-24,33; we take extreme values 132 as upper limit and -24 as lower limit]
So C is not True. Answer is C
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