(x^2+4)(x^2+4x)>0

Question

Find and graph the solution set of the inequality

Answer 1

Aloha Wendy,

 

The two factors you have in parentheses are both parabolas going toward +∞ as +-X gets very large.

The (x² + 4) factor has a minimum point (x,y) = (0, +4) so it's always positive for all x.

The other factor (x² + 4x) has a minimum at (x,y) = (-2, -4) and is zero at x=0 and x=-4.

 

Now in general, the product of two numbers can only be strictly positive if neither one is zero and both are either positive or negative simultaneously.

 

The first factor is always positive and never zero, so we can exclude the (-)(-) = (+) option here.  But the second factor can be zero or negative over the [-4 <= x <= 0] range.  Hence, we have to exclude this interval from the solution which holds for all other +- x values.