To solve the inequality \( \frac{2}{5}m - 7 \leq 17 \), we will justify each step using properties of inequalities.
Step 1: Isolate the term with \( m \)
Start by adding \( 7 \) 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: Adding the same number (in this case, \( 7 \)) to both sides of an inequality maintains the inequality. This is known as the Addition Property of Inequality.
Step 2: Solve for \( m \)
Next, multiply both sides of the inequality by \( \frac{5}{2} \):
\[ \frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2} \]
This simplifies to:
\[ m \leq 60 \]
Justification: Multiplying both sides of an inequality by a positive number (in this case, \( \frac{5}{2} \)) maintains the direction of the inequality. This is known as the Multiplication Property of Inequality.
Thus, the final solution to the inequality is:
\[ m \leq 60 \]
These steps showed how we used properties of inequalities to justify our operations while solving for \( m \).
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