Start with a graph and write the inequality from the graph.
Inequality: 52≤ s≤ 58 Graph:
Practice makes perfect
We are told that the cruise-control system should keep the speed within 3 miles per hour of the speed set to 55 mph. 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 is the key word "within." These tell us that we have an "and" compound inequality, whose solution set is the overlap between the lesser value and the greater value.
Lesser value: &55-3&= 52mph
Greater value: &55+3&= 58ppm
Finally, the word "within" 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, 52mph. 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 52mph is less than or equal to the speed range.
52≤ s
The solution set also lies to the left of the greater point, 58mph, and this point is also closed. This means that the speed range is less than or equal to 58mph.
s≤ 58
These two statements form an "and" compound inequality.
52≤ sand x≤ 58 ⇔ 52≤ s≤ 58
