1. **State the problem:** Find the equation of the tangent line to the curve $f(x) = \frac{7}{x - 2}$ at $x = 3$. 2. **Recall the formula:** The equation of the tangent line at $x = a$ is given by: $$y = f(a) + f'(a)(x - a)$$ where $f'(a)$ is the derivative of $f(x)$ evaluated at $x = a$. 3. **Find $f(3)$:** $$f(3) = \frac{7}{3 - 2} = \frac{7}{1} = 7$$ 4. **Find the derivative $f'(x)$:** Rewrite $f(x)$ as $7(x - 2)^{-1}$. Using the power rule and chain rule: $$f'(x) = 7 \cdot (-1)(x - 2)^{-2} \cdot 1 = -\frac{7}{(x - 2)^2}$$ 5. **Evaluate $f'(3)$:** $$f'(3) = -\frac{7}{(3 - 2)^2} = -\frac{7}{1^2} = -7$$ 6. **Write the tangent line equation:** $$y = f(3) + f'(3)(x - 3) = 7 - 7(x - 3)$$ Simplify: $$y = 7 - 7x + 21 = -7x + 28$$ **Final answer:** The equation of the tangent line at $x=3$ is $$y = -7x + 28$$