Solve for \( x \) (1) \( \frac{3}{x-1}>0 \) 2) \( \frac{-2}{x+3} \geq 0 \)

Solve the equation \( \frac{3}{x-1}>0 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(\frac{3}{x-1}>0\) - step1: Find the domain: \(\frac{3}{x-1}>0,x\neq 1\) - step2: Set the denominator of \(\frac{3}{x-1}\) equal to 0\(:\) \(x=1\) - step3: Determine the test intervals: \(\begin{align}&x<1\\&x>1\end{align}\) - step4: Choose a value: \(\begin{align}&x_{1}=0\\&x_{2}=2\end{align}\) - step5: Test the chosen value: \(\begin{align}&x<1\textrm{ }\textrm{is not a solution}\\&x>1\textrm{ }\textrm{is the solution}\end{align}\) - step6: The final solution is \(x>1:\) \(x>1\) - step7: Check if the solution is in the defined range: \(x>1,x\neq 1\) - step8: Find the intersection: \(x>1\) Solve the equation \( \frac{-2}{x+3} \geq 0 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(\frac{-2}{x+3}\geq 0\) - step1: Find the domain: \(\frac{-2}{x+3}\geq 0,x\neq -3\) - step2: Rewrite the fraction: \(-\frac{2}{x+3}\geq 0\) - step3: Change the sign: \(\frac{2}{x+3}\leq 0\) - step4: Set the denominator of \(\frac{2}{x+3}\) equal to 0\(:\) \(x=-3\) - step5: Determine the test intervals: \(\begin{align}&x<-3\\&x>-3\end{align}\) - step6: Choose a value: \(\begin{align}&x_{1}=-4\\&x_{2}=-2\end{align}\) - step7: Test the chosen value: \(\begin{align}&x<-3\textrm{ }\textrm{is the solution}\\&x>-3\textrm{ }\textrm{is not a solution}\end{align}\) - step8: Include the critical value: \(\begin{align}&x<-3\textrm{ }\textrm{is the solution}\end{align}\) - step9: The final solution is \(x<-3:\) \(x<-3\) - step10: Check if the solution is in the defined range: \(x<-3,x\neq -3\) - step11: Find the intersection: \(x<-3\) The solutions to the given inequalities are: 1) \( x > 1 \) 2) \( x < -3 \)