To solve the inequality \( 25m - 7 \leq 17 \), let's go through the steps and justify each of them using properties of inequalities.
Step 1: Isolate the term with \( m \)
We want to get \( 25m \) by itself on one side of the inequality. To do this, we'll 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 property states that if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \).
Step 2: Solve for \( m \)
Next, we divide both sides of the inequality by 25 to isolate \( m \).
\[ \frac{25m}{25} \leq \frac{24}{25} \]
This simplifies to:
\[ m \leq \frac{24}{25} \]
Justification: Dividing both sides of an inequality by a positive number (in this case, 25) also does not change the direction of the inequality. This property states that if \( a \leq b \) and \( c > 0 \), then \( \frac{a}{c} \leq \frac{b}{c} \).
Putting it all together, the solution to the inequality \( 25m - 7 \leq 17 \) is:
\[ m \leq \frac{24}{25} \]
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