Simplifying each side of the equation results in \( x^{2}-3 x-4=x^{2}-5 x+6 \) Find the solution: \( \frac{x+2}{3 x}-\frac{1}{x-2}=\frac{x-3}{3 x} \)

Solve the equation \( \frac{x+2}{3x}-\frac{1}{x-2}=\frac{x-3}{3x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x+2}{3x}-\frac{1}{x-2}=\frac{x-3}{3x}\) - step1: Find the domain: \(\frac{x+2}{3x}-\frac{1}{x-2}=\frac{x-3}{3x},x \in \left(-\infty,0\right)\cup \left(0,2\right)\cup \left(2,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{x+2}{3x}-\frac{1}{x-2}\right)\times 3x\left(x-2\right)=\frac{x-3}{3x}\times 3x\left(x-2\right)\) - step3: Simplify the equation: \(x^{2}-3x-4=x\left(x-2\right)-3\left(x-2\right)\) - step4: Calculate: \(x^{2}-3x-4=x^{2}-2x-3x+6\) - step5: Simplify: \(x^{2}-3x-4=x^{2}-5x+6\) - step6: Cancel equal terms: \(-3x-4=-5x+6\) - step7: Move the expression to the left side: \(-3x-4-\left(-5x+6\right)=0\) - step8: Subtract the terms: \(2x-10=0\) - step9: Move the constant to the right side: \(2x=0+10\) - step10: Remove 0: \(2x=10\) - step11: Divide both sides: \(\frac{2x}{2}=\frac{10}{2}\) - step12: Divide the numbers: \(x=5\) - step13: Check if the solution is in the defined range: \(x=5,x \in \left(-\infty,0\right)\cup \left(0,2\right)\cup \left(2,+\infty\right)\) - step14: Find the intersection: \(x=5\) The solution to the equation \( \frac{x+2}{3x}-\frac{1}{x-2}=\frac{x-3}{3x} \) is \( x=5 \).