Simple Inequality Principle

Question

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!

BrushMyQuant · Answer

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!

KarishmaB · Answer

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.

Simple Inequality Principle

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