Content - Solving Systems of Quadratic Inequalities

How do we normally schedule a picnic?

Solving Systems of Quadratic Inequalities

What do you do if you want to schedule a picnic among a few friends? One way to do it is to talk to each individual friend and find out when he/she is available. After you talk to all of them, put all schedules together and see if there is any time when everybody is available. If the answer is yes, we find the solution. Otherwise, we may have to cancel the picnic.

As shown, the 17th appears to be a perfect day for the picnic.

Similarly, to solve a system of inequalities, we need to solve each individual inequality first. Then, find the overlapped area. Because this overlapped region can satisfy all the inequalities, this area is the solution to the system. If there is no overlapped area, then the solution does not exist. The following graphs demonstrate the two cases.

The overlapped area can satisfy both

y > x² - 1
and
y < -x² + 1

Thus, it is the solution to this system of quadratic inequalities.

There is no overlapped area which can satisfy both

y > x² + 1
and
y < -x² - 1

Thus, there is no solution to this system of quadratic inequalities.

[Experiment] [Exercise]