Write the word sentence as an inequality.
A number x is at most 3.
The Correct answer and Explanation is :
The word sentence “A number ( x ) is at most 3” can be written as the inequality:
[
x \leq 3
]
Explanation:
To break this down:
- What does “at most” mean?
The phrase “at most” means that the number cannot be greater than a certain value but can be less than or equal to it. In this case, the number ( x ) cannot be greater than 3 but can be 3 or any number less than 3. - Translation to a mathematical inequality:
We want to express that the number ( x ) is either less than or equal to 3. This is exactly what the inequality ( x \leq 3 ) conveys. The symbol ( \leq ) means “less than or equal to,” which matches the meaning of “at most.” - Breaking down the components of the inequality:
- ( x ) represents the unknown number we are discussing.
- The symbol ( \leq ) indicates that the number on the left (in this case ( x )) is less than or equal to the number on the right (3).
- The number 3 is the upper bound, meaning that ( x ) cannot exceed 3.
- Why not use ( x < 3 )?
Using ( x < 3 ) would imply that the number must be strictly less than 3 and cannot equal 3. However, “at most” means the number could also be 3, so we use ( \leq ), which includes the possibility that ( x ) could equal 3.
This inequality is useful for situations where there is an upper limit but no lower limit specified. For example, if you’re given a rule that says, “You can spend at most 3 hours on an activity,” this means you could spend 3 hours or any amount of time less than that, but no more.