To represent the situation "all real numbers at most –9.5 or at least 5.5" using inequalities, you can write it as:
\[ x \leq -9.5 \quad \text{or} \quad x \geq 5.5 \]
This captures that \( x \) can be any real number that is either less than or equal to -9.5 or greater than or equal to 5.5.