1. **State the problem:** Simplify the expression $x - \frac{1}{x} - \frac{1}{x} + \frac{1}{x} + \frac{1}{x}$ and solve for $x$ if possible. 2. **Simplify the expression:** Combine like terms carefully. $$x - \frac{1}{x} - \frac{1}{x} + \frac{1}{x} + \frac{1}{x} = x + \left(-\frac{1}{x} - \frac{1}{x} + \frac{1}{x} + \frac{1}{x}\right)$$ Inside the parentheses, the terms cancel out: $$-\frac{1}{x} - \frac{1}{x} + \frac{1}{x} + \frac{1}{x} = 0$$ So the expression simplifies to: $$x$$ 3. **Interpretation:** The simplified expression is just $x$. 4. **Regarding the other expressions:** - The equation $x - \frac{1}{x} = -x - x$ can be rewritten as: $$x - \frac{1}{x} = -2x$$ Add $2x$ to both sides: $$x + 2x - \frac{1}{x} = 0 \implies 3x - \frac{1}{x} = 0$$ Multiply both sides by $x$ (assuming $x \neq 0$): $$3x^2 - 1 = 0$$ Solve for $x^2$: $$3x^2 = 1 \implies x^2 = \frac{1}{3}$$ Take square roots: $$x = \pm \frac{1}{\sqrt{3}}$$ 5. **Final answer:** The simplified expression is $x$. If solving the equation $x - \frac{1}{x} = -2x$, the solutions are: $$x = \pm \frac{1}{\sqrt{3}}$$