Algebra
# Floor and Ceiling Functions

Which of the following statements is/are true?

I. $\lceil x+n\rceil=\lceil x\rceil+n$ for any real $x$ and any integer $n.$

II. $\lceil x\rceil+\lceil-x\rceil=1$ if and only if $x$ is not an integer.

III. $\lceil x\rceil+\lceil y\rceil\le\lceil x+y\rceil\le\lceil x\rceil+\lceil y\rceil+1$ for any real $x$ and $y.$

What is the solution set to $\lceil\frac{x}{3}-8\rceil=10?$

What is $\lceil\log_{5}59\rceil?$

How many integers $x$ satisfy the equation $\lceil\log_{3}x\rceil=4?$

What is $\lceil-\pi\rceil?$