The problem
gives practice in working with linear inequalities and in working
systematically through separate cases.
Possible approach
Encourage the class to try some numerical values for $x$, to
compare the values of the three functions and to record their
findings. Collect sufficient results from the class to provide
evidence for spotting patterns and making conjectures.
Key questions
If the integer part of $x$ is $a$ then $x=a + b$ where $a$ is a
whole number and $0\leq b < 1$. What is the difference between
the separate cases where $0 \leq b < {1\over 2}$ and ${1\over
2}\leq b < 1$?