1. **State the problem:** Solve the inequality $$\frac{x - 1}{x} - x < 1 - \frac{x}{n + 1}$$ for $x$. 2. **Rewrite the inequality:** $$\frac{x - 1}{x} - x < 1 - \frac{x}{n + 1}$$ 3. **Find a common denominator and simplify terms:** First, rewrite each term clearly: - Left side: $$\frac{x - 1}{x} - x = \frac{x - 1}{x} - \frac{x^2}{x} = \frac{x - 1 - x^2}{x} = \frac{-x^2 + x - 1}{x}$$ - Right side: $$1 - \frac{x}{n + 1} = \frac{(n + 1)}{n + 1} - \frac{x}{n + 1} = \frac{n + 1 - x}{n + 1}$$ 4. **Rewrite inequality with these expressions:** $$\frac{-x^2 + x - 1}{x} < \frac{n + 1 - x}{n + 1}$$ 5. **Cross-multiply carefully, considering domain restrictions:** $$(-x^2 + x - 1)(n + 1) < x(n + 1 - x)$$ 6. **Expand both sides:** Left side: $$(-x^2 + x - 1)(n + 1) = -x^2(n + 1) + x(n + 1) - (n + 1)$$ Right side: $$x(n + 1 - x) = x(n + 1) - x^2$$ 7. **Bring all terms to one side:** $$-x^2(n + 1) + x(n + 1) - (n + 1) - x(n + 1) + x^2 < 0$$ Simplify terms: $$-x^2(n + 1) + x^2 - (n + 1) < 0$$ 8. **Factor the $x^2$ terms:** $$x^2(- (n + 1) + 1) - (n + 1) < 0$$ Simplify inside parentheses: $$x^2(-n - 1 + 1) - (n + 1) < 0$$ $$x^2(-n) - (n + 1) < 0$$ 9. **Rewrite inequality:** $$-n x^2 - (n + 1) < 0$$ Multiply both sides by $-1$ (reverse inequality): $$n x^2 + (n + 1) > 0$$ 10. **Analyze the inequality:** Since $x^2 \geq 0$ for all real $x$, and $n x^2 \geq 0$ if $n \geq 0$, then: - If $n \geq 0$, then $n x^2 + (n + 1) \geq n + 1 > 0$ (assuming $n + 1 > 0$), so inequality holds for all $x$. - If $n < 0$, the expression may be negative for some $x$. 11. **Domain restrictions:** - $x \neq 0$ (denominator in original inequality) - $n + 1 \neq 0$ (denominator in original inequality) 12. **Final conclusion:** The inequality simplifies to $$n x^2 + (n + 1) > 0$$ with domain restrictions $x \neq 0$ and $n \neq -1$. The solution depends on the value of $n$: - For $n \geq 0$, the inequality holds for all $x \neq 0$. - For $n < 0$, solve $$n x^2 + (n + 1) > 0$$ accordingly. **Answer:** The inequality reduces to $$n x^2 + (n + 1) > 0$$ with $x \neq 0$ and $n \neq -1$.