1. **State the problem:** Solve the equation $f(|x|) = 0$ where $f(x) = 3|x - 2| - 10$. 2. **Write the equation:** $$f(|x|) = 3\left| |x| - 2 \right| - 10 = 0$$ 3. **Isolate the absolute value term:** $$3\left| |x| - 2 \right| = 10$$ 4. **Divide both sides by 3:** $$\cancel{3}\left| |x| - 2 \right| = \frac{10}{\cancel{3}}$$ $$\left| |x| - 2 \right| = \frac{10}{3}$$ 5. **Solve the absolute value equation:** $$|x| - 2 = \frac{10}{3} \quad \text{or} \quad |x| - 2 = -\frac{10}{3}$$ 6. **Solve each case:** - Case 1: $$|x| = 2 + \frac{10}{3} = \frac{6}{3} + \frac{10}{3} = \frac{16}{3}$$ - Case 2: $$|x| = 2 - \frac{10}{3} = \frac{6}{3} - \frac{10}{3} = -\frac{4}{3}$$ 7. **Interpret the results:** Since $|x|$ cannot be negative, discard $|x| = -\frac{4}{3}$. 8. **Solve for $x$ from $|x| = \frac{16}{3}$:** $$x = \frac{16}{3} \quad \text{or} \quad x = -\frac{16}{3}$$ **Final answer:** $$x = \pm \frac{16}{3}$$