\( \left. \begin{array} { l } { f ( x ) = g ( x ) } \\ { \frac { - 3 } { x } - 2 = - x - 2 } \end{array} \right. \)

Solve the equation \( \frac{-3}{x}-2=-x-2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{-3}{x}-2=-x-2\) - step1: Find the domain: \(\frac{-3}{x}-2=-x-2,x\neq 0\) - step2: Cancel equal terms: \(\frac{-3}{x}=-x\) - step3: Cross multiply: \(-3=x\left(-x\right)\) - step4: Simplify the equation: \(-3=-x^{2}\) - step5: Evaluate: \(3=x^{2}\) - step6: Swap the sides: \(x^{2}=3\) - step7: Simplify the expression: \(x=\pm \sqrt{3}\) - step8: Separate into possible cases: \(\begin{align}&x=\sqrt{3}\\&x=-\sqrt{3}\end{align}\) - step9: Check if the solution is in the defined range: \(\begin{align}&x=\sqrt{3}\\&x=-\sqrt{3}\end{align},x\neq 0\) - step10: Find the intersection: \(\begin{align}&x=\sqrt{3}\\&x=-\sqrt{3}\end{align}\) - step11: Rewrite: \(x_{1}=-\sqrt{3},x_{2}=\sqrt{3}\) The solutions to the equation \( \frac{-3}{x}-2=-x-2 \) are \( x = -\sqrt{3} \) and \( x = \sqrt{3} \).