To solve the inequality
\[ \frac{5(x - 6)}{3} \geq \frac{3x}{5} - 15, \]
we will first clear the fractions by finding a common multiple. The common denominator for 3 and 5 is 15.
Multiply every term by 15:
\[ 15 \cdot \frac{5(x - 6)}{3} \geq 15 \cdot \left( \frac{3x}{5} - 15 \right) \]
This simplifies to:
\[ 5 \cdot 5(x - 6) \geq 3 \cdot 15x - 15 \cdot 15. \]
Simplifying further, we get:
\[ 25(x - 6) \geq 45x - 225. \]
Distributing \(25\) on the left side,
\[ 25x - 150 \geq 45x - 225. \]
Now, we can rearrange the terms to isolate \(x\). Move \(25x\) to the right side and \(-225\) to the left side:
\[ -150 + 225 \geq 45x - 25x. \]
This simplifies to:
\[ 75 \geq 20x. \]
Now, divide both sides by \(20\):
\[ \frac{75}{20} \geq x. \]
This simplifies to:
\[ \frac{15}{4} \geq x, \]
or equivalently:
\[ x \leq \frac{15}{4}. \]
Thus, the solution to the inequality is:
\[ x \leq \frac{15}{4}. \]
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