Combining Inequalities: Finding the Solution Set for Quadratic Inequalities

[t_n_p]

I want to find value for m for which:

4m² - 12m > 0

Say I do this algebraically:

4m(m-3) > 0

so m > 0 or m > 3

The answer however is 0 < m and m > 3, I know this as a fact as I have looked graphically.

So, my question is, when done algebraically, how do I get 0 < m instead of m > 0 ?

 

[tiny-tim]

hi t_n_p!

4m(m-3) > 0

so m > 0 or m > 3

nooo …

if AB is positive, then either both or neither are positive …

in this case either m > 0 and m > 3 (ie m > 3), or m < 0 and m < 3 (ie m < 0)

 

[t_n_p]

ah ok, so you need to take into account the 2 conditions.

if A = 4m and B = m-3

i.e. condition 1)
A>0 & B>0 yields m>0 and m>3

but m>3 is the overriding (is there a better word to use?) condition

condition 2)
A<0 and B<0 yields m<0 and m<3

but m<0 is the overriding condition

so then I combine the conditions to yield m<0 and m>3

Ok, makes sense now, thanks!

 

[Mentallic]

Yes it's called the intersection of the two inequalities
When you require both inequalities to hold, you say m<0 AND m<3, which leaves the intersection of the two, m<0.

 

[Mark44]

Another way to think of it is this.
If m > 0 AND m > 3, then any number m larger than 3 is automatically larger than 0, so saying m > 0 is redundant. Note however, that a number m that is positive is not necessarily larger than 3.

 

[Hurkyl]

so then I combine the conditions to yield m<0 and m>3

As an aside, I would like to point out an awkwardness of language. The two following statements are different:

• The solution set is the union of the set described by "m<0" or the set described by "m>3".
• The solution set is described by the system of inequalities "m<0 and m>3"

Although the English phrase you used could be interpreted either way. (You surely meant the first one)