A compound inequality describes a range with two limits. For a tank draining at a constant rate, multiply the rate by the time to find how much leaves, then subtract from the starting volume. Evaluate the two rate boundaries in Desmos and check whether each one counts. Don't copy the rate's inequality signs onto the volume: faster draining leaves less water.
Hints
- Hint 1
Let be the rate in gallons drained per minute. A greater than limit leaves its boundary out, while an at most limit includes its boundary. Write both conditions together before finding any volumes.
- Hint 2
At a constant rate of gallons per minute, the tank loses gallons in minutes. The question asks for water remaining, so what should you do with that loss?
- Hint 3
Evaluate the two boundary rates in the remaining-volume rule. The larger rate leaves less water. For each resulting volume, ask whether the rate that produced it is allowed.
Step-by-step
Check the boundary rates in Desmos
Step 1Mark the allowed rates
Let be the rate in gallons drained per minute. “Greater than” excludes , while “at most” includes , so:
- Step 2
Model the water remaining
In minutes, the tank loses gallons. The remaining volume is the starting gallons minus that loss:
A higher draining rate leaves less water.
- Step 3
Find the included minimum volume
The largest allowed rate is , so it leaves the least water. Type , where is the rate, then type into Desmos. It shows . This volume is included because the rate is allowed.
- Step 4
Find the excluded maximum volume
The lower rate boundary is . Below the earlier Desmos lines, type . It shows . But the rate must be greater than , so is not allowed and gallons is excluded.
- Step 5
Write the range of remaining volumes
As the rate increases steadily from above through , every volume between the boundaries occurs. The compound inequality for the gallons remaining is:
Choice A.