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If \(1 > 1 - ab > 0,\) which of the following must be true?

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by Vincen » Thu Dec 09, 2021 11:44 am
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If \(1 > 1 - ab > 0,\) which of the following must be true?

I. \(\dfrac{a}{b} > 0\)

I. \(\dfrac{a}{b} < 1\)

III. \(ab < 1\)

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Answer: E

Source: Manhattan GMAT
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Re: If \(1 > 1 - ab > 0,\) which of the following must be true?

by Brent@GMATPrepNow » Fri Dec 10, 2021 2:06 pm
Vincen wrote: ↑
Thu Dec 09, 2021 11:44 am
 If \(1 > 1 - ab > 0,\) which of the following must be true?

I. \(\dfrac{a}{b} > 0\)

I. \(\dfrac{a}{b} < 1\)

III. \(ab < 1\)

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Answer: E

Source: Manhattan GMAT
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.

Answer: E
Brent Hanneson - Creator of GMATPrepNow.com
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