A truck driver is driving from Nome, Alaska, to Death Valley, California. Because he is traveling between locations with extreme temperatures, he needs to check the weather continuously to make sure the gas in his truck remains in liquid form. The gas he uses freezes at −40° F and evaporates at 140° F.
Part A: Write an inequality to represent the temperatures at which the gas in the truck will remain in liquid form.
Part B: Describe the graph of the inequality completely from Part A. Use terms such as open/closed circles and shading directions. Explain what the solutions to the inequality represent.
Part C: In January 1989, the temperature in Nome, Alaska, dropped to −49° F. Would the gas in the driver's truck have remained in liquid form so he could have driven on this day? Why or why not?
Part A:
To represent the temperatures at which the gas in the truck will remain in liquid form, we need an inequality that includes the range of temperatures between −40° F and 140° F, not including the endpoints where the gas freezes or evaporates. The inequality that represents this situation is:
-40° F < Temperature < 140° F
This inequality states that the Temperature must be greater than −40° F but less than 140° F for the gas to remain in liquid form.
Part B:
To graph this inequality on a number line, we would draw a number line and represent the interval between −40° F and 140° F with two points on the line. Since −40° F and 140° F are not included in the solutions (the gas changes state at these temperatures), we would use open circles to denote these points. The open circles indicate that the endpoints are not included in the solution set. We would then shade the region between the two open circles to represent all the temperatures that are solutions to the inequality.
The graph would look something like this:
```
Number line: -------------------------------------------------
-50 -40 0 100 140 150
Points/circles: ( ) ( )
Shading: <-------------->
```
The solutions to the inequality represent all the possible temperatures at which the gas in the truck remains in liquid form, which is the desired state for the truck to operate properly.
Part C:
In January 1989, the temperature in Nome, Alaska, dropped to −49° F. At this temperature, the inequality −40° F < Temperature < 140° F is not satisfied, because −49° F is less than −40° F. Therefore, the gas in the driver's truck would not have remained in liquid form; it would have frozen. Consequently, under these conditions, the driver would not have been able to drive on this day.