To solve the inequality \( 25m - 7 \leq 17 \), we proceed with the following steps:

To solve the inequality \( 25m - 7 \leq 17 \), we can follow these steps:

### Step 1: Add 7 to both sides

By adding the same value to both sides, we maintain the direction of the inequality:

\[
25m - 7 + 7 \leq 17 + 7
\]

This simplifies to:

\[
25m \leq 24
\]

### Step 2: Divide both sides by 25

Since we are dividing both sides by a positive value, the inequality remains unchanged:

\[
\frac{25m}{25} \leq \frac{24}{25}
\]

This simplifies to:

\[
m \leq \frac{24}{25}
\]

### Conclusion

Therefore, the solution to the inequality \( 25m - 7 \leq 17 \) is:

\[
m \leq \frac{24}{25}
\]