Bunuel
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.1 > 1 - ab > 0
Multiply by -1.......1*(-1)<ab-1<0 => -1<ab-1<0
Add 1 to each part of inequality.....0<ab<1
This means a and b have the same sign, and at least one of the two is a fraction between 0 and |-1|.
Let us look at the options.
I. a/b > 0
This tells us that a and b have the same sign. We may not know the value of a and b, but surely they have the same sign and, therefore, a/b>0.
II. a/b < 1
Now, we do not know the value of a and b. we just know that at least of a and b is a fraction between 1 and -1.
Say a=1/3 and b=2.......\(\frac{a}{b}=\frac{\frac{1}{3}}{2}=\frac{1}{6}\)
Say b=1/3 and a=2.......\(\frac{a}{b}=\frac{2}{\frac{1}{3}}=\frac{2*3}{1}=6\)
Need not be true always.
III. ab < 1
We have already seen the inequality mean 0<ab<1.
True
I and III
E