To solve the inequality \( \frac{5}{6}x \leq -10 \), we need to isolate \( x \).
First, we can multiply both sides of the inequality by \( \frac{6}{5} \) to eliminate the fraction, while keeping in mind that when we multiply or divide both sides of an inequality by a positive number, the direction of the inequality symbol remains the same.
So,
\[
x \leq -10 \times \frac{6}{5}
\]
Calculating the right side, we get:
\[
x \leq -12
\]
Now, examining each person's claim:
- **Juana** is incorrect. The inequality symbol is not reversed just because there is a negative sign; it only needs to be reversed if you multiply or divide by a negative number.
- **Anish** is correct. The division or multiplication properties can apply here, and we multiplied by a positive number \( \frac{6}{5} \), so the inequality remains unchanged.
- **Giselle** is also correct in saying the inequality symbol must stay the same since we multiplied by a positive number.
Therefore, the correct combination of people is **Anish and Giselle**.