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

To solve the inequality \( 25m - 7 \leq 17 \), we will go through the steps with proper justifications using properties of inequalities.

**Step 1: Add 7 to both sides.**

Starting with the original inequality:
\[
25m - 7 \leq 17
\]

We add 7 to both sides:
\[
25m - 7 + 7 \leq 17 + 7
\]
This simplifies to:
\[
25m \leq 24
\]

**Justification:** The property used here is that **adding the same number to both sides of an inequality does not change the direction of the inequality**. In mathematical terms, if \( a \leq b \) and \( c \) is any real number, then \( a + c \leq b + c \).

---

**Step 2: Divide both sides by 25.**

Now we have:
\[
25m \leq 24
\]

We divide both sides by 25:
\[
\frac{25m}{25} \leq \frac{24}{25}
\]
This simplifies to:
\[
m \leq \frac{24}{25}
\]

**Justification:** The property used here is that **dividing both sides of an inequality by a positive number does not change the direction of the inequality**. In mathematical terms, if \( a \leq b \) and \( c > 0 \), then \( \frac{a}{c} \leq \frac{b}{c} \).

---

In summary, by adding 7 and dividing by 25 (both positive numbers), we solved the inequality while maintaining its truth. The solution to the inequality \( 25m - 7 \leq 17 \) is:
\[
m \leq \frac{24}{25}
\]