To solve the inequality \( \frac{2}{5} m - 7 \leq 17 \), we will perform two key steps, and I'll justify both using properties of inequalities.
Step 1: Add 7 to both sides
We start with the inequality:
\[ \frac{2}{5} m - 7 \leq 17 \]
To isolate the term containing \( m \), we 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: Adding the same value (in this case, \( +7 \)) to both sides of an inequality preserves the inequality’s direction. Therefore, since both sides of the inequality were increased by the same amount, the relationship remains valid.
Step 2: Multiply both sides by \( \frac{5}{2} \)
Next, we want to solve for \( m \) by eliminating the fraction \( \frac{2}{5} \). To do this, we multiply both sides of the inequality by \( \frac{5}{2} \):
\[ \left( \frac{5}{2} \times \frac{2}{5} \right) m \leq 24 \times \frac{5}{2} \]
This simplifies to:
\[ m \leq 60 \]
Justification: When we multiply or divide both sides of an inequality by a positive number (in this case, \( \frac{5}{2} \)), the direction of the inequality remains the same. Hence, multiplying both sides by \( \frac{5}{2} \) does not change the inequality sign.
Final Answer
The final solution to the inequality is:
\[ m \leq 60 \]
Both steps were justified using the relevant properties of inequalities.
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