Step 1: Recall modulation index.
In amplitude modulation, a message signal rides on a carrier wave. The modulation index $m$ measures how strongly the message changes the carrier's amplitude.
Step 2: Write its meaning.
The modulation index is the ratio of the message signal amplitude to the carrier amplitude: $m = \dfrac{A_m}{A_c}$.
Step 3: Look at the healthy range.
When $m$ is between 0 and 1, the carrier amplitude stays positive at all times and the signal is recovered cleanly without distortion.
Step 4: Look at what happens above 1.
If $m$ grows larger than 1, the message amplitude is bigger than the carrier amplitude. The envelope of the wave then crosses zero and folds over.
Step 5: Name this condition.
This folding distorts the signal and is called over-modulation. The recovered message no longer matches the original.
Step 6: State the condition.
So over-modulation happens when the modulation index is greater than 1, that is $m > 1$. \[ \boxed{m > 1} \]