Use algebraic and /or graphical methods to solve the inequality

Question

m^3-4m>0

Answer 1

m^3-4m>0

 

Factored, the inequality is:  m(m+2)(m-2) > 0

 

Now, consider what values of the three factors make the inequality true.  They are:

   1.   (+)(+)(+)

   2.   (+)(-)(-)

   3.    (-)(+)(-)

   4.    (-)(-)(+)

All other combinations produce values < 0.  Note that 0 is neither + or -.

 

So,

  [1]  m>0 and m>-2 and m>2    means that m>2

  [2]  m>0 and m<-2 and m<2     isn't possible

  [3]  m<0 and m>-2 and m<2    means that -2<m<0

  [4]  m<0 and m<-2 and m>2     isn't possible

 

The solution is:   -2<m<0 or m>2

Answer 2

m^3-4m = m(m^2 - 4) = m(m+2)(m-2)

 

Multiplying with the following combination of signs of each factor will keep the product greater than zero.

 

m(m+2)(m-2) > 0

(+)  (+)  ( +)    

 

m>0  m>-2    m>2             =====> m > 2 is common for the three inequalities that multiply to meet the requirement

 

(+)   (-)    (-)

 

m>0   m<-2   m<2             ======> nothing in common for all inequalities

 

(-)    (-)    (+)

 

m<0  m<-2  m>2              =======> nothing in common for all inequalities

 

(-)    (+)    (-)

 

m<0   m>-2  m<2              =======>   -2<m<0

 

 

So, the answer that holds for the inequality in question is:

 

m>2  or -2<m<0

 

You can use a number line to verify this answer.

 

Answer 3

You can factor out an M and you get m(m²-4)>0

The (m²-4)>0 can be factored to (m-2)(m+2)>0.  (Remember a²-b²=(a-b)(a+b))

Now set all factors to be greater than zero and solve.

 

m>0   m-2>0  and m+2>0

 

so m>0, m>2, and m>-2