﻿ Practice with the phrases "at least" and "at most"
PRACTICE WITH THE PHRASES ‘AT LEAST’ AND ‘AT MOST’
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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 $\,5\,$
Master the ideas from this section