To solve the inequality \( \frac{2}{5}m - 7 \leq 17 \), we will justify both steps using properties of inequalities.
Step 1: Isolate the term with \( m \)
Action: Add 7 to both sides of the inequality.
\[ \frac{2}{5}m - 7 + 7 \leq 17 + 7 \]
This simplifies to:
\[ \frac{2}{5}m \leq 24 \]
Justification: When we add the same value (in this case, 7) to both sides of an inequality, the direction of the inequality remains the same. This is a property of inequalities known as the "Addition Property of Inequality.”
Step 2: Solve for \( m \)
Action: Multiply both sides of the inequality by \( \frac{5}{2} \).
\[ \left(\frac{5}{2}\right) \cdot \frac{2}{5}m \leq 24 \cdot \left(\frac{5}{2}\right) \]
This simplifies to:
\[ m \leq 60 \]
Justification: When multiplying both sides of an inequality by a positive number (in this case, \( \frac{5}{2} \)), the direction of the inequality remains the same. This is known as the "Multiplication Property of Inequality."
Conclusion
Combining the justified steps, we have:
- \( \frac{2}{5}m - 7 \leq 17 \) → Add 7: \( \frac{2}{5}m \leq 24 \)
- Multiply by \( \frac{5}{2} \): \( m \leq 60 \)
Thus, the solution to the inequality is:
\[ m \leq 60 \]