Using the properties of inequality, justify both steps in solving the inequality 25m−7≤17 .

To solve the inequality \( 25m - 7 \leq 17 \), we can break down the steps and justify each one using the properties of inequalities.

### Initial Inequality
1. **Start with the inequality**:
\[
25m - 7 \leq 17
\]

### Step 1: Add 7 to both sides
2. **Justification**: According to the **Addition Property of Inequality**, if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \). Here, we are adding 7 to both sides of the inequality to isolate the term with \( m \).
\[
25m - 7 + 7 \leq 17 + 7
\]
This simplifies to:
\[
25m \leq 24
\]

### Step 2: Divide both sides by 25
3. **Justification**: According to the **Division Property of Inequality**, if \( a \leq b \) and \( c > 0 \), then \( \frac{a}{c} \leq \frac{b}{c} \). Here, \( c = 25 \) (which is greater than 0). Thus, we can divide both sides of the inequality by 25.
\[
\frac{25m}{25} \leq \frac{24}{25}
\]
This simplifies to:
\[
m \leq \frac{24}{25}
\]

### Final Solution
4. **Conclusion**: The solution to the inequality \( 25m - 7 \leq 17 \) is:
\[
m \leq \frac{24}{25}
\]

These steps justify the manipulation of the inequality, demonstrating the use of properties of inequalities for valid operations throughout the solution.