To solve the inequality \( 25m - 7 \leq 17 \), we can break it down into two steps and justify each step using properties of inequalities.
**Step 1: Add 7 to both sides**
We start by isolating the term involving \( m \). We add 7 to both sides of the inequality:
\[
25m - 7 + 7 \leq 17 + 7
\]
This simplifies to:
\[
25m \leq 24
\]
**Justification:** Adding the same number (in this case, 7) to both sides of an inequality does not change the direction of the inequality. This is a property of inequalities that states: if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \).
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**Step 2: Divide both sides by 25**
Next, we divide both sides of the inequality by 25:
\[
\frac{25m}{25} \leq \frac{24}{25}
\]
This simplifies to:
\[
m \leq \frac{24}{25}
\]
**Justification:** Dividing both sides of an inequality by a positive number (in this case, 25) does not change the direction of the inequality. This is another property of inequalities which states: if \( a \leq b \) and \( c > 0 \), then \( \frac{a}{c} \leq \frac{b}{c} \).
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Combining both steps, the solution to the inequality \( 25m - 7 \leq 17 \) is:
\[
m \leq \frac{24}{25}
\]