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Re: If 1 > 1 - ab > 0, which of the following must be true?
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24 Feb 2018, 07:25
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Top Contributor
Bunuel wrote:
If 1 > 1 - ab > 0, which of the following must be true?
I. a/b > 0 I. a/b < 1 III. ab < 1
(A) I only (B) II only (C) III only (D) I and II only (E) I and III only
Kudos for a correct solution.
GIVEN: 1 > 1 - ab > 0 Multiply all 3 sides by -1 to get: -1 < -1 + ab < 0 [since I multiplied by a NEGATIVE number, I had to REVERSE the inequality symbols] Add 1 to all 3 sides to get: 0 < ab < 1
First, if 0 < ab, then a and b are the SAME SIGN, which means a/b > 0 So, statement I is TRUE ELIMINATE B and C
Second, our new inequality clearly tells us that ab < 1 So, statement III is TRUE ELIMINATE A and D
So, we need not even check whether statement II is true.
Re: If 1 > 1 - ab > 0, which of the following must be true?
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24 Jun 2015, 01:36
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I am going to go with E.
1>1-ab>0; If we split this into two separate equations (1-ab<1 & 1-ab>0), we get the following solutions - ab>0 & ab<1.
I says that a/b>0. Since ab>0, it means that either a,b both are positive or both are negative. In either scenario I holds true.
III says that ab<1. We already got that result be splitting and solving the inequality.
II says that a/b<1. Let's take two cases - Case 1 (a=2, b=0.2) & Case 2 (a=0.2, b=2). II holds true for Case 2 but not Case 1, and therefore cannot always be true.
If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
Updated on: 24 Jun 2015, 09:15
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Step 1: Subtracting 1 from all, 0>-ab>-1. Step 2: Multiplying by -1, signs are reversed. Therefore we get : 0<ab<1 Therefore, we can conclude : 1. both ab have the same sign 2. ab lies between 0 to 1
Hence E !!
Originally posted by NickHalden on 24 Jun 2015, 01:38.
Last edited by NickHalden on 24 Jun 2015, 09:15, edited 1 time in total.
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
24 Jun 2015, 06:27
3
Kudos
If 1 > 1 - ab > 0, which of the following must be true?
I. a/b > 0 I. a/b < 1 III. ab < 1
(A) I only (B) II only (C) III only (D) I and II only (E) I and III only
Solution - a. For the part 1 > 1 - ab -> Subtract -1 from both sides gives -ab < 0 -> ab > 0. Both a and b are positive or negative. b. For the part 1 - ab > 0 -> Add ab on both sides gives -> ab<1.
I. a/b > 0. This will give us both a and b are positive or negative. Meets the condition a above. Sufficient. II. a/b < 1. This is opposite of condition a above. In Sufficient. III. ab < 1. This inequality satisfy the condition b above. Sufficient.
Concentration: Technology, Social Entrepreneurship
WE:Information Technology (Computer Software)
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
24 Jun 2015, 09:09
Rearranging the 1>1-ab>0, equation we get 1>1-ab or ab>0 ---1 1-ab>0 or ab<1 -----2 A and b both has to be same sign from equation 1, so a/b will always be >0 buy in some cases it will be >2 or <1 .Hence choose 1 From equation 2 it’s clear that ab<1, hence choose 3. [This can also be solved by taking example where we have d>a>l , and cross verify ]
You can manipulate the original compound inequality as follows, making sure to perform each manipulation on every term:
1 > 1 - ab > 0
0 > -a b > -1 Subtract 1 from all three terms.
0 < ab < 1 Multiply all three terms by -1 and flip the inequality signs.
Therefore you know that 0 < ab < 1. This tells you that ab is positive, so a/b must be positive (a and b have the same sign). Therefore, I must be true. However, you do not know whether a/b < 1, so II is not necessarily true. But you do know that ab must be less than 1, so III must be true.
Therefore, the correct answer is (E). _________________
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
30 Jun 2015, 07:15
1
Kudos
Expert Reply
If 1 > 1 - ab > 0, which of the following must be true?
I. a/b > 0 I. a/b < 1 III. ab < 1
If you first deal with the right half of the inequality, you can add ab to both sides to get ab<1. Since III is true, you can eliminate A, B, and D.
Now test I. Looking at the left side of the inequality, add ab to both sides and subtract 1 from both sides to yield ab>0. For ab to be positive, a and b must have the same signs. This will also be true if we divide a by b. Statement I is also true. (A) I only (B) II only (C) III only (D) I and II only (E) I and III only
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
30 Jun 2015, 08:46
0 < 1-ab < 1, since (1-ab) is both positive and less than 1, 0<ab<1, so III is true. For (ab) to be less than 1 and greater than 0 the fraction a/b must be greater than 0 (must be positive), so I is also true. Ans is E? _________________
I used to think the brain was the most important organ. Then I thought, look what’s telling me that.
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
15 Jul 2015, 08:19
Hello Bunuel , While solving this problem , i reached up till this inequality ( 0<ab<1) . However , statement III above says that ab<1 . This covers a lot many more numbers, which wont be satisfied by the inequality provided in the stem ( e.g since ab<1 , it will also mean that ab =-3). Should this not be reason good enough to eliminate statement III (as this has provided us with values that are unable to satisfy the stem) ?
PS: If the question had asked , "which of the following might be true ?", then,yes, we could still include statement III. I am unable to understand this. Could you please help in this ? Thanks in advance.
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
15 Jul 2015, 08:27
Expert Reply
fetchnitin wrote:
Hello Bunuel , While solving this problem , i reached up till this inequality ( 0<ab<1) . However , statement III above says that ab<1 . This covers a lot many more numbers, which wont be satisfied by the inequality provided in the stem ( e.g since ab<1 , it will also mean that ab =-3). Should this not be reason good enough to eliminate statement III (as this has provided us with values that are unable to satisfy the stem) ?
PS: If the question had asked , "which of the following might be true ?", then,yes, we could still include statement III. I am unable to understand this. Could you please help in this ? Thanks in advance.
We have that 0 < ab < 1. Now, let me asks you is ab < 1 true? _________________
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
15 Jul 2015, 08:56
Hello, yes,this is true .ab<1 . But my confusion is that it also contains values that are outside the range of the inequality in the question stem . Is it that, while solving these type of questions all the values of the question stem should be tried and checked with the options and not the other way round ( i.e all values of the options should fit the range of stem ). If so ,then could u explain to me the logic behind this, as i am unable to visualize this .
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
15 Jul 2015, 09:00
Hello Bunuel, yes,this is true .ab<1 . But my confusion is that it also contains values that are outside the range of the inequality in the question stem . Is it that, while solving these type of questions all the values of the question stem should be tried and checked with the options and not the other way round ( i.e all values of the options should fit the range of stem ). If so ,then could u explain to me the logic behind this, as i am unable to visualize this .
Re: If 1 > 1 - ab > 0, which of the following must be true?
[#permalink]
15 Jul 2015, 09:03
Expert Reply
fetchnitin wrote:
Hello Bunuel, yes,this is true .ab<1 . But my confusion is that it also contains values that are outside the range of the inequality in the question stem . Is it that, while solving these type of questions all the values of the question stem should be tried and checked with the options and not the other way round ( i.e all values of the options should fit the range of stem ). If so ,then could u explain to me the logic behind this, as i am unable to visualize this .
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