Aim: how do we solve quadratic inequalities ? 

To solve quadratic inequality, first you can either graph or factor the inequality to see where the graph hits the x-axis. Then test different points to see which points make the inequality true. 

For example,
y< x^2-x-12  
First, factor or graph it to find where the parabola hits the x-axis 


y< x^2-x-12  
y = x^2-x-12  

Zero Product Property:
0= x^2-x-12  
0 =(x+3)(x-4)

x+3=0                         x-4=0
  -3  -3                           +4 +4   
x= -3                              x=4

Now, to find out where the points will make the inequality true, pick test points from both inside and outside the graph to test.


Inside:
(0,0)
0< (0)^2-(0)-12
0< 0-0-12
0< -12              FALSE


Outside: 
(-4,-8)
-8< (-4)^2-(-4)-12
-8< 16+4-12
-8< 20-12
-8< 8                   TRUE


Finally, the solution is: x<-2 or x>4 


Now that you know the points outside the parabola makes the graph true, you shade the outside of the parabola.