An audio engineer finds that her mixes sound best when the reverb () is less than or equal to this expression: and the reverb is more than or equal to this expression: , where represents the number of instruments in the mix. These expressions can be written as inequalities:
Here’s a graph** of these two inequalities:
a. Explain what steps you would take to find the points of intersection of the two curves on the graph
b. Determine which of the following points lie in the optimal range:
(8, 80), (16, 100), (22, 80 ), (20, 80), (16, 60), (19.5, 73.6)
c. How do you know which of these points is in the optimal range when you use the graph?
d. How would you prove which of these points are in the optimal range by using algebra?
**For your convenience, you can access a more interactive graph here. You can use the magnification tools on the graph to zoom (though the interactive graph does not have any shading for the inequalities).