1. **State the problem:** Solve the inequality $5 - \frac{2}{x} < 1$ for $x$. 2. **Rewrite the inequality:** $$5 - \frac{2}{x} < 1$$ 3. **Isolate the fraction:** $$5 - 1 < \frac{2}{x}$$ $$4 < \frac{2}{x}$$ 4. **Rewrite the inequality as:** $$\frac{2}{x} > 4$$ 5. **Multiply both sides by $x^2$ (which is always positive) to avoid reversing inequality sign:** $$2x > 4x^2$$ 6. **Bring all terms to one side:** $$0 > 4x^2 - 2x$$ $$0 > 2x(2x - 1)$$ 7. **Analyze the product $2x(2x - 1)$:** - The product is less than zero when the factors have opposite signs. 8. **Find intervals:** - $2x > 0 \Rightarrow x > 0$ - $2x - 1 < 0 \Rightarrow x < \frac{1}{2}$ 9. **Combine intervals:** $$0 < x < \frac{1}{2}$$ 10. **Check domain restrictions:** - $x \neq 0$ because of denominator. **Final answer:** $$\boxed{0 < x < \frac{1}{2}}$$