Question 1209113: Find all values of x such that
\frac{2x}{x + 2} = \frac{x - 3}{x - 4} + \frac{2x + 7}{x + 1}.
If you find more than one value, then list your solutions, separated by commas.
Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website! To solve this equation, we'll first clear the denominators by multiplying both sides by $(x+2)(x-4)(x+1)$:
$$2x(x-4)(x+1) = (x-3)(x+1)(x+2) + (2x+7)(x-4)(x+2)$$
Expand both sides:
$$2x^3 - 6x^2 - 8x = x^3 - 8x - 3 + 2x^3 - 2x^2 - 22x - 56$$
Combine like terms:
$$-x^2 - 12x - 59 = 0$$
This is a quadratic equation. We can solve it using the quadratic formula:
$$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(-1)(-59)}}{2(-1)}$$
Simplifying:
$$x = \frac{12 \pm \sqrt{-100}}{-2}$$
Since the discriminant is negative, there are no real solutions to this equation.
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