Hi,

The solution requires that you know the definition of absolute value.

\[ |z| = \left\{ \begin{array}{rl} z & \mbox{if $z \ge 0$} \\ -z & \mbox{if $z < 0$}\end{array}\right. \]

Thus for your problem if $|x - 3| \ge 0$, that is if $x \ge 3$ then the expression is

\[ 5 - |x - 3| = 5 - (x - 3) = 5 - x + 3 = 8 - x\]

What is the greatest value of $8 - x$ for $x$ an integral and $x \ge 3?$

Now suppose $|x - 3| < 0.$ What does your expression become?

Penny