Question 136·Easy·Linear Inequalities in One or Two Variables
A refrigerated warehouse must keep its internal temperature no lower than and no higher than . Which inequality represents the acceptable temperature range, where is the temperature (in degrees Celsius) inside the warehouse?
For word problems about temperature or other measurements staying within a range, first translate each verbal condition into its own inequality: phrases like "no lower than" or "at least" mean ≥, and "no higher than" or "at most" mean ≤. Then combine the two inequalities into a single compound inequality by putting the variable in the middle, the smaller number on the left, and the larger number on the right with ≤ signs. Finally, match this compound inequality to the choice that shows both bounds correctly and in order.
Hints
Focus on the lower limit
Look at the phrase "no lower than -2°C." Ask yourself: Is -3°C allowed? What about -2°C or 0°C? Decide whether T must be greater than, less than, or equal to -2°C.
Now look at the upper limit
Consider the phrase "no higher than 4°C." Is 5°C allowed? Is 4°C allowed? Decide whether T must be greater than, less than, or equal to 4°C.
Combine your two inequalities
Once you have one inequality for the lower bound and one for the upper bound, think about how to write both conditions together in a single statement with T in the middle and the smaller number on the left.
Desmos Guide
Graph the lower bound inequality
In Desmos, type x >= -2. This will shade the region where x is greater than or equal to -2. (You can use x instead of T; the choice of variable name does not change the inequality.)
Graph the upper bound inequality
On a new line, type x <= 4. This shades the region where x is less than or equal to 4.
Find the overlapping region
Look at the graph and focus on where the two shaded regions overlap along the x-axis. Note the smallest and largest x-values in this overlap, and then write a single compound inequality that describes all x-values in that overlapping region.
Step-by-step Explanation
Translate "no lower than -2°C"
The warehouse must keep its temperature no lower than -2°C.
- If the temperature were -3°C, that would be lower than -2°C, so it would NOT be allowed.
- Temperatures of -2°C or warmer (like -1°C, 0°C, 1°C, etc.) are allowed.
This means the temperature T must be greater than or equal to -2, which we write as .
Translate "no higher than 4°C"
The warehouse must keep its temperature no higher than 4°C.
- If the temperature were 5°C, that would be higher than 4°C, so it would NOT be allowed.
- Temperatures of 4°C or colder (like 3°C, 2°C, 0°C, etc.) are allowed.
This means the temperature T must be less than or equal to 4, which we write as .
Combine both conditions into one inequality
The temperature must satisfy both conditions at the same time:
- (at least -2°C)
- (at most 4°C)
We can combine these into a single compound inequality by putting T in the middle, with the smaller number on the left and the larger number on the right:
This matches answer choice C.