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PRACTICE WITH THE PHRASES "AT LEAST" AND "AT MOST"

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The concepts for this exercise are summarized below.
For a complete discussion, read the text. (Click here for solutions to the text exercises.)


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 ≥ 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 ≤ 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 ≥ 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 ≤ k

EXAMPLES:
Suppose that  x  must take 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  
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Suppose that  x  must take on values between   and   .
Then,


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