Lewis Daniel
12/15/2023 · Senior High School
1. \( \frac{5 x}{2 x+1}+\frac{3 x+1}{3 x-5}=6 \)
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Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( \frac{5x}{2x+1}+\frac{3x+1}{3x-5}=6 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{5x}{2x+1}+\frac{3x+1}{3x-5}=6\)
- step1: Find the domain:
\(\frac{5x}{2x+1}+\frac{3x+1}{3x-5}=6,x \in \left(-\infty,-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\frac{5}{3}\right)\cup \left(\frac{5}{3},+\infty\right)\)
- step2: Multiply both sides of the equation by LCD:
\(\left(\frac{5x}{2x+1}+\frac{3x+1}{3x-5}\right)\left(2x+1\right)\left(3x-5\right)=6\left(2x+1\right)\left(3x-5\right)\)
- step3: Simplify the equation:
\(21x^{2}-20x+1=36x^{2}-42x-30\)
- step4: Move the expression to the left side:
\(21x^{2}-20x+1-\left(36x^{2}-42x-30\right)=0\)
- step5: Calculate:
\(-15x^{2}+22x+31=0\)
- step6: Multiply both sides:
\(15x^{2}-22x-31=0\)
- step7: Solve using the quadratic formula:
\(x=\frac{22\pm \sqrt{\left(-22\right)^{2}-4\times 15\left(-31\right)}}{2\times 15}\)
- step8: Simplify the expression:
\(x=\frac{22\pm \sqrt{\left(-22\right)^{2}-4\times 15\left(-31\right)}}{30}\)
- step9: Simplify the expression:
\(x=\frac{22\pm \sqrt{2344}}{30}\)
- step10: Simplify the expression:
\(x=\frac{22\pm 2\sqrt{586}}{30}\)
- step11: Separate into possible cases:
\(\begin{align}&x=\frac{22+2\sqrt{586}}{30}\\&x=\frac{22-2\sqrt{586}}{30}\end{align}\)
- step12: Simplify the expression:
\(\begin{align}&x=\frac{11+\sqrt{586}}{15}\\&x=\frac{22-2\sqrt{586}}{30}\end{align}\)
- step13: Simplify the expression:
\(\begin{align}&x=\frac{11+\sqrt{586}}{15}\\&x=\frac{11-\sqrt{586}}{15}\end{align}\)
- step14: Check if the solution is in the defined range:
\(\begin{align}&x=\frac{11+\sqrt{586}}{15}\\&x=\frac{11-\sqrt{586}}{15}\end{align},x \in \left(-\infty,-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\frac{5}{3}\right)\cup \left(\frac{5}{3},+\infty\right)\)
- step15: Find the intersection:
\(\begin{align}&x=\frac{11+\sqrt{586}}{15}\\&x=\frac{11-\sqrt{586}}{15}\end{align}\)
- step16: Rewrite:
\(x_{1}=\frac{11-\sqrt{586}}{15},x_{2}=\frac{11+\sqrt{586}}{15}\)
La solución de la ecuación \( \frac{5x}{2x+1}+\frac{3x+1}{3x-5}=6 \) es \( x_{1}=\frac{11-\sqrt{586}}{15} \) y \( x_{2}=\frac{11+\sqrt{586}}{15} \).
Quick Answer
La solución de la ecuación es \( x_{1}=\frac{11-\sqrt{586}}{15} \) y \( x_{2}=\frac{11+\sqrt{586}}{15} \).
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