How do I solve inequalities when they're not linear?

Ok, let's talk about quadratic inequalities in particular. Look at the inequality x^2 - 8x + 15 > 0.What does a solution to this mean? It means the curve x^2 - 8x + 15 > 0 is above the x-axis at this point. So let's start by finding where the curve actually intersects the x-axis. That is, solutions to x^2 - 8x + 15 = 0.We know how to do this: just factorise. We get (x-5)(x-3) = 0, so the curve intersects the axis at x = 5 and at x = 3.Now, if you think about the shape of the curve, you will see that it is a parabola with a minimum and no maximum, since the x^2 term is positive. So in fact it must be that the curve is below the x-axis between x = 3 and x = 5, and above the axis everywhere else.So the solution to the inequality is x < 3 or x > 5.