Solve by graphing: \( \frac{x+9}{2 x}=x+2 \) \( \left\{-\frac{3}{2}, 3\right\} \) \( \left\{-3, \frac{3}{2}\right\} \) \( \{-3,2\} \) (-3)

Solve the equation \( \frac{x+9}{2x}=x+2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x+9}{2x}=x+2\) - step1: Find the domain: \(\frac{x+9}{2x}=x+2,x\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\frac{x+9}{2x}\times 2x=\left(x+2\right)\times 2x\) - step3: Simplify the equation: \(x+9=2x^{2}+4x\) - step4: Move the expression to the left side: \(x+9-\left(2x^{2}+4x\right)=0\) - step5: Calculate: \(-3x+9-2x^{2}=0\) - step6: Factor the expression: \(\left(3+x\right)\left(3-2x\right)=0\) - step7: Separate into possible cases: \(\begin{align}&3+x=0\\&3-2x=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=-3\\&x=\frac{3}{2}\end{align}\) - step9: Check if the solution is in the defined range: \(\begin{align}&x=-3\\&x=\frac{3}{2}\end{align},x\neq 0\) - step10: Find the intersection: \(\begin{align}&x=-3\\&x=\frac{3}{2}\end{align}\) - step11: Rewrite: \(x_{1}=-3,x_{2}=\frac{3}{2}\) The solutions to the equation \( \frac{x+9}{2x}=x+2 \) are \( x = -3 \) and \( x = \frac{3}{2} \). Therefore, the solutions are \( x = -3 \) and \( x = \frac{3}{2} \), which corresponds to the option \( \{-3, \frac{3}{2}\} \).