1. **State the problem:** Solve the equation $1+\left|1+\frac{1}{x}\right|=4$ for $x$. 2. **Isolate the absolute value:** Subtract 1 from both sides: $$\left|1+\frac{1}{x}\right|=4-1=3$$ 3. **Remove the absolute value:** The expression inside the absolute value can be either 3 or -3: $$1+\frac{1}{x}=3 \quad \text{or} \quad 1+\frac{1}{x}=-3$$ 4. **Solve each case separately:** **Case 1:** $$1+\frac{1}{x}=3$$ Subtract 1: $$\frac{1}{x}=3-1=2$$ Invert both sides: $$x=\frac{1}{2}$$ **Case 2:** $$1+\frac{1}{x}=-3$$ Subtract 1: $$\frac{1}{x}=-3-1=-4$$ Invert both sides: $$x=\frac{1}{-4}=-\frac{1}{4}$$ 5. **Final answer:** $$x=\frac{1}{2} \quad \text{or} \quad x=-\frac{1}{4}$$