5 Resuelve las siguientes ecuaciones: a) \( \frac{1}{x}-\frac{1}{x+3}=\frac{3}{10} \) b) \( \frac{4}{x}+\frac{2(x+1)}{3(x-2)}=4 \) c) \( \frac{1}{x}+\frac{1}{x^{2}}=\frac{3}{4} \)

Solve the equation \( \frac{1}{x}-\frac{1}{x+3}=\frac{3}{10} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{1}{x}-\frac{1}{x+3}=\frac{3}{10}\) - step1: Find the domain: \(\frac{1}{x}-\frac{1}{x+3}=\frac{3}{10},x \in \left(-\infty,-3\right)\cup \left(-3,0\right)\cup \left(0,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{1}{x}-\frac{1}{x+3}\right)\times 10x\left(x+3\right)=\frac{3}{10}\times 10x\left(x+3\right)\) - step3: Simplify the equation: \(30=3x^{2}+9x\) - step4: Swap the sides: \(3x^{2}+9x=30\) - step5: Move the expression to the left side: \(3x^{2}+9x-30=0\) - step6: Factor the expression: \(3\left(x-2\right)\left(x+5\right)=0\) - step7: Divide the terms: \(\left(x-2\right)\left(x+5\right)=0\) - step8: Separate into possible cases: \(\begin{align}&x-2=0\\&x+5=0\end{align}\) - step9: Solve the equation: \(\begin{align}&x=2\\&x=-5\end{align}\) - step10: Check if the solution is in the defined range: \(\begin{align}&x=2\\&x=-5\end{align},x \in \left(-\infty,-3\right)\cup \left(-3,0\right)\cup \left(0,+\infty\right)\) - step11: Find the intersection: \(\begin{align}&x=2\\&x=-5\end{align}\) - step12: Rewrite: \(x_{1}=-5,x_{2}=2\) Solve the equation \( \frac{4}{x}+\frac{2(x+1)}{3(x-2)}=4 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{4}{x}+\frac{2\left(x+1\right)}{3\left(x-2\right)}=4\) - step1: Find the domain: \(\frac{4}{x}+\frac{2\left(x+1\right)}{3\left(x-2\right)}=4,x \in \left(-\infty,0\right)\cup \left(0,2\right)\cup \left(2,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{4}{x}+\frac{2\left(x+1\right)}{3\left(x-2\right)}\right)\times 3x\left(x-2\right)=4\times 3x\left(x-2\right)\) - step3: Simplify the equation: \(14x-24+2x^{2}=12x^{2}-24x\) - step4: Move the expression to the left side: \(14x-24+2x^{2}-\left(12x^{2}-24x\right)=0\) - step5: Calculate: \(38x-24-10x^{2}=0\) - step6: Factor the expression: \(-2\left(3-x\right)\left(4-5x\right)=0\) - step7: Divide the terms: \(\left(3-x\right)\left(4-5x\right)=0\) - step8: Separate into possible cases: \(\begin{align}&3-x=0\\&4-5x=0\end{align}\) - step9: Solve the equation: \(\begin{align}&x=3\\&x=\frac{4}{5}\end{align}\) - step10: Check if the solution is in the defined range: \(\begin{align}&x=3\\&x=\frac{4}{5}\end{align},x \in \left(-\infty,0\right)\cup \left(0,2\right)\cup \left(2,+\infty\right)\) - step11: Find the intersection: \(\begin{align}&x=3\\&x=\frac{4}{5}\end{align}\) - step12: Rewrite: \(x_{1}=\frac{4}{5},x_{2}=3\) Solve the equation \( \frac{1}{x}+\frac{1}{x^{2}}=\frac{3}{4} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{1}{x}+\frac{1}{x^{2}}=\frac{3}{4}\) - step1: Find the domain: \(\frac{1}{x}+\frac{1}{x^{2}}=\frac{3}{4},x\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{1}{x}+\frac{1}{x^{2}}\right)\times 4x^{2}=\frac{3}{4}\times 4x^{2}\) - step3: Simplify the equation: \(4x+4=3x^{2}\) - step4: Move the expression to the left side: \(4x+4-3x^{2}=0\) - step5: Factor the expression: \(\left(2-x\right)\left(2+3x\right)=0\) - step6: Separate into possible cases: \(\begin{align}&2-x=0\\&2+3x=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=2\\&x=-\frac{2}{3}\end{align}\) - step8: Check if the solution is in the defined range: \(\begin{align}&x=2\\&x=-\frac{2}{3}\end{align},x\neq 0\) - step9: Find the intersection: \(\begin{align}&x=2\\&x=-\frac{2}{3}\end{align}\) - step10: Rewrite: \(x_{1}=-\frac{2}{3},x_{2}=2\) Las soluciones de las ecuaciones dadas son: a) \( x = -5 \) o \( x = 2 \) b) \( x = \frac{4}{5} \) o \( x = 3 \) c) \( x = -\frac{2}{3} \) o \( x = 2 \)