To solve the inequality
\[ \frac{5(x - 6)}{3} \geq \frac{3x}{5} - 15, \]
we'll start by eliminating the fractions. The least common multiple of the denominators \(3\) and \(5\) is \(15\). We will multiply the entire inequality by \(15\) to clear the fractions:
\[ 15 \cdot \frac{5(x - 6)}{3} \geq 15 \left(\frac{3x}{5} - 15\right). \]
Calculating each side:
Left side: \[ 15 \cdot \frac{5(x - 6)}{3} = 5 \cdot 5(x - 6) = 25(x - 6) = 25x - 150. \]
Right side: \[ 15 \left(\frac{3x}{5} - 15\right) = 3 \cdot 3x - 15 \cdot 15 = 9x - 225. \]
Now we have the inequality:
\[ 25x - 150 \geq 9x - 225. \]
Next, we'll move the terms involving \(x\) to one side and the constant terms to the other side. Subtract \(9x\) from both sides:
\[ 25x - 9x - 150 \geq -225. \] \[ 16x - 150 \geq -225. \]
Now, add \(150\) to both sides:
\[ 16x \geq -225 + 150. \] \[ 16x \geq -75. \]
Now, divide both sides by \(16\):
\[ x \geq \frac{-75}{16}. \]
Thus, the solution to the inequality is:
\[ x \geq -\frac{75}{16}. \]
In decimal form, \(-\frac{75}{16} \approx -4.6875\). Therefore, the final solution can be expressed as:
\[ x \geq -4.6875. \]