To solve the inequality \( \frac{2}{5}m - 7 \leq 17 \), we can follow these steps while justifying each step using properties of inequalities:
Step 1: Add 7 to both sides.
Justification: When you add the same value to both sides of an inequality, the inequality sign remains the same.
Starting with the original inequality:
\[ \frac{2}{5}m - 7 \leq 17 \]
Add 7 to both sides:
\[ \frac{2}{5}m - 7 + 7 \leq 17 + 7 \]
This simplifies to:
\[ \frac{2}{5}m \leq 24 \]
Step 2: Multiply both sides by \( \frac{5}{2} \).
Justification: When you multiply both sides of an inequality by a positive number, the inequality sign remains the same.
Multiply both sides by \( \frac{5}{2} \):
\[ \frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2} \]
The left side simplifies as follows:
\[ m \leq \frac{120}{2} \]
This simplifies to:
\[ m \leq 60 \]
Conclusion: The solution to the inequality \( \frac{2}{5}m - 7 \leq 17 \) is \( m \leq 60 \). Both steps maintain the inequality correctly, as justified above.
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