Practice With The Phrases "at Least" And "at Most"

Concept Discussion Example Skill Practice Full Text (pdf) 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 $\,5\,$

Practice

Two different numbers appear in each question. Choose one of these numbers as your answer! Your choice must make the given sentence true for every member of the specified set.