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 \( c \).
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 \( \frac{a}{c} \leq \frac{b}{c} \).
Combining both steps, the solution to the inequality \( 25m - 7 \leq 17 \) is:
\[ m \leq \frac{24}{25} \]