### Why do this
problem?

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 ${1\over
2}\leq b < 1$?