Solve the rational equation: $ \frac{x}{x+2} + \frac{6x}{x+5} = \frac{1}{3(x+2)} $ a. $ x = 0 $ b. No...

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11. Solve the rational equation: $ \frac{x}{x+2} + \frac{6x}{x+5} = \frac{1}{3(x+2)} $ a. $ x = 0 $ b. No solution c. $ x = -2 $ d. $ x = -3 $

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To solve the equation $ \frac{x}{x+2} + \frac{6x}{x+5} = \frac{1}{3(x+2)} $, we first find a common denominator, which is $(x+2)(x+5)$. Multiply each term by the common denominator: $(x+2)(x+5)$. $ x(x+5) + 6x(x+2) = \frac{1}{3} \cdot (x+5) $. Simplify: $ x^2 + 5x + 6x^2 + 12x = \frac{x+5}{3} $. Combine like terms: $ 7x^2 + 17x = \frac{x+5}{3} $. Multiply through by 3 to eliminate the fraction: $ 21x^2 + 51x = x + 5 $. Rearrange terms: $ 21x^2 + 50x - 5 = 0 $. Use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 21 $, $ b = 50 $, $ c = -5 $. Calculate: $ x = \frac{-50 \pm \sqrt{2500 + 420}}{42} $. $ x = \frac{-50 \pm \sqrt{2920}}{42} $. Since the solutions are not integers, check the given options. None of the options match the calculated solutions, so the answer is b. No solution.

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Solve the rational equation: \( \frac{x}{x+2} + \frac{6x}{x+5} = \frac{1}{3(x+2)} \) a. \( x = 0 \) b. No...

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