BACK TO TEXT

2.4 Introduction:  Solving Quadratic Inequalities

Recall the real-world scenario discussed in Section 2.1 .  It is reprinted below for your convenience. 

Suppose that you run a Music Store and the revenue in thousands of dollars from the sale of x  thousand CDs is given by .   If the cost in thousands of dollars, of producing n  thousand CDs is given by , how many CDs do you need to sell in order to break even?

Recall that to answer the above question, we set the function equal to 0 and solved it.  Now

suppose we wanted to know when the revenue was  negative, i.e. when would the business be going in the hole?  We need to find the exact point where the revenue changes from positive to negative.  In other words, first we need to find where the revenue is zero.  This is the problem we already solved in Section 2.1.    We set the function = 0 and solved it:

Factor:    

Set each factor = 0:      so   so .

Thus we had two values of n for which the Revenue was 0, n = 0 or n =  . This means that when we sell 0 CDs, we have 0 revenue, which makes sense.  If we sell  thousand CDs, (-500 of them)  we have 0 revenue.  This negative value for n does not make sense in the context of the problem, so we can ignore it.

     The graph of the revenue function is below, enlarged around the origin.  Though to solve the real world problem, we would probably ignore negative values of n, let’s take a look at the graph divorced from its real world context.  

Notice that the output is negative  for  or, written as an interval: .

  In the following section, we will be solving problems like these, i.e. we will be looking for intervals for which the output of a quadratic function is positive or negative .  We call these types of problems quadratic inequalities.

BACK TO TEXT