To solve the inequality \( \frac{2}{5}m - 7 \leq 17 \), we will perform two steps and provide justification for each step using properties of inequalities.
Step 1: Add 7 to both sides
Inequality: \[ \frac{2}{5}m - 7 \leq 17 \]
Justification: Adding the same number (in this case, 7) to both sides of an inequality preserves the direction of the inequality. This property states that if \( a \leq b \), then \( a + c \leq b + c \) for any real number \( c \).
New Inequality: \[ \frac{2}{5}m - 7 + 7 \leq 17 + 7 \]
Simplifying: \[ \frac{2}{5}m \leq 24 \]
Step 2: Multiply both sides by \( \frac{5}{2} \)
Inequality: \[ \frac{2}{5}m \leq 24 \]
Justification: When multiplying both sides of an inequality by a positive number, the direction of the inequality remains unchanged. This is a fundamental property of inequalities: if \( a \leq b \) and \( c > 0 \), then \( ac \leq bc \).
New Inequality: \[ \frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2} \]
Simplifying: On the left side, the \( \frac{5}{2} \) and \( \frac{2}{5} \) cancel out, leaving \( m \): \[ m \leq 60 \]
Conclusion:
The solution to the original inequality \( \frac{2}{5}m - 7 \leq 17 \) is \( m \leq 60 \). Each step followed proper properties of inequalities, ensuring the integrity of the solution.
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