Find the solution for the rational equation: \( ((2 /(x+1)+5 / 2 x)=2 \backslash) \) Answer:

Solve the equation \( \frac{2}{x+1}+\frac{5}{2x}=2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{2}{x+1}+\frac{5}{2x}=2\) - step1: Find the domain: \(\frac{2}{x+1}+\frac{5}{2x}=2,x \in \left(-\infty,-1\right)\cup \left(-1,0\right)\cup \left(0,+\infty\right)\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{2}{x+1}+\frac{5}{2x}\right)\times 2x\left(x+1\right)=2\times 2x\left(x+1\right)\) - step3: Simplify the equation: \(9x+5=4x^{2}+4x\) - step4: Move the expression to the left side: \(9x+5-\left(4x^{2}+4x\right)=0\) - step5: Calculate: \(5x+5-4x^{2}=0\) - step6: Rewrite in standard form: \(-4x^{2}+5x+5=0\) - step7: Multiply both sides: \(4x^{2}-5x-5=0\) - step8: Solve using the quadratic formula: \(x=\frac{5\pm \sqrt{\left(-5\right)^{2}-4\times 4\left(-5\right)}}{2\times 4}\) - step9: Simplify the expression: \(x=\frac{5\pm \sqrt{\left(-5\right)^{2}-4\times 4\left(-5\right)}}{8}\) - step10: Simplify the expression: \(x=\frac{5\pm \sqrt{105}}{8}\) - step11: Separate into possible cases: \(\begin{align}&x=\frac{5+\sqrt{105}}{8}\\&x=\frac{5-\sqrt{105}}{8}\end{align}\) - step12: Check if the solution is in the defined range: \(\begin{align}&x=\frac{5+\sqrt{105}}{8}\\&x=\frac{5-\sqrt{105}}{8}\end{align},x \in \left(-\infty,-1\right)\cup \left(-1,0\right)\cup \left(0,+\infty\right)\) - step13: Find the intersection: \(\begin{align}&x=\frac{5+\sqrt{105}}{8}\\&x=\frac{5-\sqrt{105}}{8}\end{align}\) - step14: Rewrite: \(x_{1}=\frac{5-\sqrt{105}}{8},x_{2}=\frac{5+\sqrt{105}}{8}\) The solutions for the rational equation \( \frac{2}{x+1}+\frac{5}{2x}=2 \) are \( x_{1}=\frac{5-\sqrt{105}}{8} \) and \( x_{2}=\frac{5+\sqrt{105}}{8} \).