Resolver las siguientes ecuaciones 1) \( \frac{3 x}{x-5}-\frac{4}{x+6}=3 \) 5) \( \frac{0,2 x+1,6}{\sqrt{1,8 \cdot 5^{-1}}}=\frac{3,3}{0,4^{-1}} \) 2) \( x\left(\frac{2 x+2}{2}\right)=50-3 x+2 \cdot(x+3)+x \) 3) \( \frac{2 x+1}{6}-\frac{x-3}{3}=\frac{4 x-1}{3}+\frac{x-6}{2} \) 4) \( \frac{3 x^{3}-18 x^{2}+27 x}{x^{3}-9 x}=1 \)

Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{3x^{3}-18x^{2}+27x}{x^{3}-9x}=1\) - step1: Find the domain: \(\frac{3x^{3}-18x^{2}+27x}{x^{3}-9x}=1,x \in \left(-\infty,-3\right)\cup \left(-3,0\right)\cup \left(0,3\right)\cup \left(3,+\infty\right)\) - step2: Divide the terms: \(\frac{3x-9}{x+3}=1\) - step3: Cross multiply: \(3x-9=x+3\) - step4: Move the expression to the left side: \(3x-x=3+9\) - step5: Add and subtract: \(2x=3+9\) - step6: Add and subtract: \(2x=12\) - step7: Divide both sides: \(\frac{2x}{2}=\frac{12}{2}\) - step8: Divide the numbers: \(x=6\) - step9: Check if the solution is in the defined range: \(x=6,x \in \left(-\infty,-3\right)\cup \left(-3,0\right)\cup \left(0,3\right)\cup \left(3,+\infty\right)\) - step10: Find the intersection: \(x=6\) Solve the equation \( \frac{0.2 x+1.6}{\sqrt{1.8 \cdot 5^{-1}}}=\frac{3.3}{0.4^{-1}} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{0.2x+1.6}{\sqrt{1.8\times 5^{-1}}}=\frac{3.3}{0.4^{-1}}\) - step1: Simplify: \(\frac{1}{3}x+\frac{8}{3}=\frac{33}{25}\) - step2: Move the constant to the right side: \(\frac{1}{3}x=\frac{33}{25}-\frac{8}{3}\) - step3: Subtract the numbers: \(\frac{1}{3}x=-\frac{101}{75}\) - step4: Multiply by the reciprocal: \(\frac{1}{3}x\times 3=-\frac{101}{75}\times 3\) - step5: Multiply: \(x=-\frac{101}{25}\) Solve the equation \( x\left(\frac{2 x+2}{2}\right)=50-3 x+2 \cdot(x+3)+x \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(x\left(\frac{2x+2}{2}\right)=50-3x+2\left(x+3\right)+x\) - step1: Divide the terms: \(x\left(x+1\right)=56\) - step2: Expand the expression: \(x^{2}+x=56\) - step3: Move the expression to the left side: \(x^{2}+x-56=0\) - step4: Factor the expression: \(\left(x-7\right)\left(x+8\right)=0\) - step5: Separate into possible cases: \(\begin{align}&x-7=0\\&x+8=0\end{align}\) - step6: Solve the equation: \(\begin{align}&x=7\\&x=-8\end{align}\) - step7: Rewrite: \(x_{1}=-8,x_{2}=7\) Solve the equation \( \frac{2 x+1}{6}-\frac{x-3}{3}=\frac{4 x-1}{3}+\frac{x-6}{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{2x+1}{6}-\frac{x-3}{3}=\frac{4x-1}{3}+\frac{x-6}{2}\) - step1: Multiply both sides of the equation by LCD: \(\left(\frac{2x+1}{6}-\frac{x-3}{3}\right)\times 6=\left(\frac{4x-1}{3}+\frac{x-6}{2}\right)\times 6\) - step2: Simplify the equation: \(7=11x-20\) - step3: Swap the sides: \(11x-20=7\) - step4: Move the constant to the right side: \(11x=7+20\) - step5: Add the numbers: \(11x=27\) - step6: Divide both sides: \(\frac{11x}{11}=\frac{27}{11}\) - step7: Divide the numbers: \(x=\frac{27}{11}\) Solve the equation \( \frac{3 x}{x-5}-\frac{4}{x+6}=3 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{3x}{x-5}-\frac{4}{x+6}=3\) - step1: Find the domain: \(\frac{3x}{x-5}-\frac{4}{x+6}=3,x \in \left(-\infty,-6\right)\cup \left(-6,5\right)\cup \left(5,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{3x}{x-5}-\frac{4}{x+6}\right)\left(x-5\right)\left(x+6\right)=3\left(x-5\right)\left(x+6\right)\) - step3: Simplify the equation: \(3x^{2}+14x+20=3x^{2}+3x-90\) - step4: Cancel equal terms: \(14x+20=3x-90\) - step5: Move the expression to the left side: \(14x-3x=-90-20\) - step6: Add and subtract: \(11x=-90-20\) - step7: Add and subtract: \(11x=-110\) - step8: Divide both sides: \(\frac{11x}{11}=\frac{-110}{11}\) - step9: Divide the numbers: \(x=-10\) - step10: Check if the solution is in the defined range: \(x=-10,x \in \left(-\infty,-6\right)\cup \left(-6,5\right)\cup \left(5,+\infty\right)\) - step11: Find the intersection: \(x=-10\) Aquí están las soluciones para las ecuaciones que proporcionaste: 1) Para la ecuación \( \frac{3 x}{x-5}-\frac{4}{x+6}=3 \): - La solución es \( x = -10 \). 2) Para la ecuación \( \frac{0.2 x+1.6}{\sqrt{1.8 \cdot 5^{-1}}}=\frac{3.3}{0.4^{-1}} \): - La solución es \( x = -\frac{101}{25} \) o aproximadamente \( x \approx -4.04 \). 3) Para la ecuación \( x\left(\frac{2 x+2}{2}\right)=50-3 x+2 \cdot(x+3)+x \): - Las soluciones son \( x_{1} = -8 \) y \( x_{2} = 7 \). 4) Para la ecuación \( \frac{2 x+1}{6}-\frac{x-3}{3}=\frac{4 x-1}{3}+\frac{x-6}{2} \): - La solución es \( x = \frac{27}{11} \) o aproximadamente \( x \approx 2.45 \). 5) Para la ecuación \( \frac{3 x^{3}-18 x^{2}+27 x}{x^{3}-9 x}=1 \): - La solución es \( x = 6 \). Si necesitas más detalles sobre cómo se resolvieron estas ecuaciones, házmelo saber.