Start with a graph and write the inequality from the graph.
Inequality: 82.4≤ f≤ 659.2 Graph:
Practice makes perfect
We are told that the range of frequencies for three octaves is between 82.4 and 659.2 Hertz, inclusive. Let's break down what this means, graph it, then write a compound inequality to represent the range.
Graphing the Compound Inequality
The first thing to notice about the given information are the key words "between" and "and." These tell us that we have an "and" compound inequality, whose solution set is the overlap between the lesser value ( 82.4Hz) and the greater value ( 659.2Hz).
Finally, the word "inclusive" tells us that both the lesser value and the greater value are included in the solution set, which is shown with closed circles on the number line.
Forming the Compound Inequality
Consider the lesser point, 82.4Hz. The solution set we showed in the graph lies to the right of this point, and the circle at this point is closed. This means that 82.4Hz is less than or equal to the range of frequencies.
82.4≤ f
The solution set also lies to the left of the greater point, 659.2Hz, and this point is also closed. This means that the range of frequencies is less than or equal to 659.2Hz.
f≤ 659.2
These two statements form an "and" compound inequality.
82.4≤ fand f≤ 659.2 ⇔ 82.4≤ f≤ 659.2
