Practice with the Phrases ‘at least’ and ‘at most’
The sentence ‘$\,x\,$ is at least $\,5\,$’ means that the least $\,x\,$ is allowed to be is $\,5\,.$ It can be $\,5\,,$ or any number greater than $\,5\,.$ So, the phrase ‘$\,x\,$ is at least $\,5\,$’ means ‘$\,x\ge 5\,$’.
The sentence ‘$\,x\,$ is at most $\,10\,$’ means that the most $\,x\,$ is allowed to be is $\,10\,.$ It can be $\,10\,,$ or any number less than $\,10\,.$ So, the phrase ‘$\,x\,$ is at most $\,10\,$’ means ‘$\,x\le 10\,$’.
More generally:
| sentence | meaning of sentence | equivalent sentence |
| $x\,$ is at least $\,k$ | the least $\,x\,$ is allowed to be is $\,k\,$; it can be $\,k\,$ or any number greater than $\,k\,$ | $x\,\ge k$ |
| $x\,$ is at most $\,k$ | the most $\,x\,$ is allowed to be is $\,k\,$; it can be $\,k\,$ or any number less than $\,k\,$ | $x\,\le k$ |
Example
Suppose that $\,x\,$ takes on values between $\,-3\,$ and $\,5\,.$ Then:
- $\,x\,$ is at least $\,-3\,$
- $\,x\,$ is at most $\,5\,$
- $\,x\,$ is greater than or equal to $\,-3\,$
- $\,x\,$ is less than or equal to $\,5\,$