Problem Solving

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

I. a/b>0

II. 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

My workings:-

1>1-ab>0
1-1>1-ab-1>0-1
0>-ab>-1

From the book:-

1>1-ab>0
0>-ab>-1
0<ab<1

Answer is E.

I don't understand why my workings does not match with the one in the book. Someone, please help.

In between, can someone teach me how to use the spoiler to hide the answer.

Thank you very much.

My workings:-

1>1-ab>0
1-1>1-ab-1>0-1
0>-ab>-1

One must be aware that, If all the terms in an inequality are multiplied by a negative number, the inequality reverses.(Read here - https://www.mathsisfun.com/algebra/inequ ... lving.html). For example, 5>4. but when you multiply both sides with -1, you get -5 < -4. (Did you see how the inequality reversed from > to <)

You are absolutely correct in what you did.Starting from where you left, let me multiply all the terms in the inequation with -1.
0>-ab>-1
0*1 < -1*-ab < -1*-1
0<ab<1.
So, from the equation we know that ab<1 (Option III is a MUST BE TRUE condition) and ab>0.

If the value of ab>0, a and b can be both positive(positive*positive = positive) or both negative(negative*negative = positive).
If a,b are both positive(say 1 and 2), a/b > 0 (1/2>0)
If a,b are both negative(say -2 and -2), a/b > 0 (-2/-1 = 2/1 >0)
So, the value of a/b is always greater than 0, i.e a/b > 0(so is the value of b/a). So, Option I is a MUST BE TRUE condition, if 0<ab<1.

We also know that the value of a/b can be greater than 1(=2 if a = -2 and b = -1) or less then 1(=1/2 if a = 1 and b = 2). So, condition II is not always true.

Answer E

p.s: Here is a link - https://www.beatthegmat.com/new-spoilers-t5302.html that will help you with the usage of spoilers.