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Simple Inequality Principle

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Simple Inequality Principle [#permalink] New post 03 Mar 2013, 23:59
Hi all,

If I have an inequality: (a/b) < 0

Why can't I derive: a < 0(b) therefore a < 0?

But in equation (a/b) = 0, I can say a = 0, right?

Please help to explain. Thanks!
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Re: Simple Inequality Principle [#permalink] New post 04 Mar 2013, 00:46
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stewartlife wrote:
Hi all,

If I have an inequality: (a/b) < 0

Why can't I derive: a < 0(b) therefore a < 0?

But in equation (a/b) = 0, I can say a = 0, right?

Please help to explain. Thanks!

Hi,

a/b < 0
then you can do a<0 PROVIDED b>0
If you are not sure about the sign of b then you can solve it like this
if b <0 then a >0
and
if b >0 then a <0

In general if you have
a/b < 0 then it simply means that a and b have opposite signs( if a is +ve then b is negative and vise versa)
Also,
a/b > 0 then it simply means that a and b have the same sign.( Either both are positive or both are negative.)
Adding to this
ab < 0 means that a and b have oppostie signs( if a is +ve then b is negative and vise versa)
And
ab > 0 means that a and b have the same sign(Either both are positive or both are negative.)

Coming to a/b = 0
Here since we don't have an inequality so the sign of b doesnt matter and it simply means that a =0 (as there is no other way in which the fraction a/b =0)

Hope it helps!
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Re: Simple Inequality Principle [#permalink] New post 04 Mar 2013, 20:09
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Expert's post
stewartlife wrote:
Hi all,

If I have an inequality: (a/b) < 0

Why can't I derive: a < 0(b) therefore a < 0?

But in equation (a/b) = 0, I can say a = 0, right?

Please help to explain. Thanks!


Think about it this way:
a/b < 0 implies a/b is negative.

When will a fraction be negative? When either numerator or denominator is negative and the other is positive.
So if a/b is negative, it means either a or b (but not both) is negative.
If \(a = 4, b = -5, a/b = -(\frac{4}{5})\) (negative)
If \(a = -4, b = 5, a/b = -(\frac{4}{5})\) (negative)

So given a/b < 0, all you can say is that one and only one of a and b is negative and the other is positive. You cannot say which one is negative and which is positive.

On the other hand, a/b = 0 only when a = 0. If b = 0, then a/b is not defined.
You can cross multiply, i.e. take b to the other side, when dealing with equations.

While dealing with inequalities, you cannot cross multiply until and unless you know the sign of the variable. If you know that b is positive, then you can take it to the other side. If you know that b is negative, then you can take it to the other side but you need to flip the inequality sign. If you do not know the sign of b, you cannot take it to the other side.

I would suggest you to check out our Algebra book for a detailed discussion on basics of Inequalities.
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Re: Simple Inequality Principle   [#permalink] 04 Mar 2013, 20:09
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