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# If b ≠ 0, and 1 – a/b > 3, which of the following must be true?

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Joined: 02 Sep 2009
Posts: 45498
If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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24 Apr 2017, 04:20
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If b ≠ 0, and 1 – a/b > 3, which of the following must be true?

I. –a > 2b
II. a < 0
III. a^3/b^3 < 0

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

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Math Expert
Joined: 02 Aug 2009
Posts: 5784
Re: If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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24 Apr 2017, 08:13
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Bunuel wrote:
If b ≠ 0, and 1 – a/b > 3, which of the following must be true?

I. –a > 2b
II. a < 0
III. a^3/b^3 < 0

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

Hi....
Easy to error in inequalities with '-' sign..
$$1-\frac{a}{b}>3$$....
$$\frac{a}{b}<-2$$..

Let's see the equations..
I. –a > 2b
Do not cross multiply if you are not aware of SIGN
So -a/b>2 does not MEAN -a>2b always. Will depend on sign of b..
MUST be true ---- NO

II. a < 0
Nothing that can tell us this..
MUST be true

III. a^3/b^3 < 0
We know a/b<-2 so a/b<0..
So $$\frac{a^3}{b^3}<0$$..
Always TRUE..

Only III..
C
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Re: If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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24 Apr 2017, 12:02
Bunuel wrote:
If b ≠ 0, and 1 – a/b > 3, which of the following must be true?

I. –a > 2b
II. a < 0
III. a^3/b^3 < 0

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

1 - a/b > 3. This can only be true when a and b have different sign.

-a > 2b
a + 2b < 0
here both a and b can be -ve i.e. same sign or different as well.

a < 0
we can have a > 0 and b < 0 and the given condition would still be true. so a < 0 is not MUST BE true.

a^3/b^3 < 0
this can only happen if a and b have different sign. hence this is a must be true condition for 1 – a/b > 3.

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Re: If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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27 Apr 2017, 17:12
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Expert's post
Bunuel wrote:
If b ≠ 0, and 1 – a/b > 3, which of the following must be true?

I. –a > 2b
II. a < 0
III. a^3/b^3 < 0

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III

We are given that 1 - a/b > 3, so -a/b > 2 or a/b < -2.

Since a/b is less than -2, either a is negative and b is positive OR a is positive and b is negative.

Let’s analyze our Roman numerals:

I. –a > 2b

If we multiply both sides of -a/b > 2 by b, then -a > 2b if b is positive OR -a < 2b if b is negative. However, since we don’t know whether b is positive or negative, we can’t determine whether -a > 2b.

II. a < 0

Since we’ve mentioned a could be either positive or negative, we can’t determine whether a < 0.

III. a^3/b^3 < 0

Since a/b < -2, a/b is negative. Since a/b is negative, a^3/b^3 = (a/b)^3 is also negative.

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Re: If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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27 Apr 2017, 22:56
Great Question this one.

Here is what i did =>

=>1-A/B >3
=>-A/B >2

We are asked as to which of the following must be true.
Option 1 =>
If B>0 => -A>2B
If B<0 => -A<2B

REJECTED.

Option 2 =>
Let A=100 and B=-2

REJECTED.

Option 3 =>
As A/B <0 => A^3/B^3 will be negative too.
This is always True.

Hence III is always True.

SMASH THAT C.

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If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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02 Oct 2017, 12:18
B not 0 and a/b <-2, means a, b has different signs and absolute value of a > that of b.

Clearly C

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Re: If b ≠ 0, and 1 – a/b > 3, which of the following must be true? [#permalink]

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27 Jan 2018, 02:03
Given: 1 - (a/b) > 3
0 > 2 + (a/b)
(a + 2b)/b < 0

case 1
if b < 0, then to hold above inequality, a + 2b > 0 => a > -2b
=> -a < 2b => since b is negative, RHS is negative, a can't be negative, if it were, then -a > 2b.
so if b < 0, then a > 0

case 2
if b > 0, then to hold above inequality, a + 2b < 0 => a < -2b
=> -a > 2b => since b is +ve, RHS is positive, a can't be positive, if it were, then -a < 2b
so if b > 0, then a < 0

Let us the see options:
(1) -a > 2b (case 2), but this need not be true always, there is possibility of case 1
(2) a < 0(case 2), need not be true, there is possibility of case 2
(3) a^3/b^3 < 0 => a, b are of opposite sign => this is always true in both the cases

(C)
Re: If b ≠ 0, and 1 – a/b > 3, which of the following must be true?   [#permalink] 27 Jan 2018, 02:03
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