Rational Expressions, solving for x

[beranabus]

The equation is 5/2 = 3/1 + 2x/x+6 + 7/x+6. I found the LCD to be 2(x+6). after that I multiplied each term by the LCD and came up with:
(x+6)(5) = 2(x+6)(3) + 2(2x+1). After distributing I got: 5x+30 = 2(3x+18)+4x+2 or 5x+30 = 6x+36+4x+2. Combined like terms: 5x+30=10x+38. Sorted the junk and I end up with -5x= 8, which I know is not correct because of the back of my textbook. So I am wondering, what went wrong? The book states the correct answer is -4.

 

[JeffM]

The equation is 5/2 = 3/1 + 2x/x+6 + 7/x+6. I found the LCD to be 2(x+6). after that I multiplied each term by the LCD and came up with:
(x+6)(5) = 2(x+6)(3) + 2(2x+1). How did you get (2x + 1)? After distributing I got: 5x+30 = 2(3x+18)+4x+2 or 5x+30 = 6x+36+4x+2. Combined like terms: 5x+30=10x+38. Sorted the junk and I end up with -5x= 8, which I know is not correct because of the back of my textbook. So I am wondering, what went wrong? The book states the correct answer is -4.

Please use proper grouping symbols, but thank you for explaining what you have done. I am guessing that the problem really is

\(\displaystyle \dfrac{5}{2} = 3 + \dfrac{2x}{x + 6} + \dfrac{7}{x + 6},\ not\ \dfrac{5}{2} = 3 + \dfrac{2x}{x} + 6 + \dfrac{7}{x} + 6,\) which is what you wrote.

\(\displaystyle \dfrac{5}{2} = 3 + \dfrac{2x}{x + 6} + \dfrac{7}{x + 6} \implies 2(x + 6) * \dfrac{5}{2} = 2(x + 6)\left(3 + \dfrac{2x}{x + 6} + \dfrac{7}{x + 6}\right) \implies\)

\(\displaystyle 5x + 30 = 2 * 3 * (x + 6) + 2 * 2x + 2 * 7 = 6x + 36 + 4x + 14 \implies\)

\(\displaystyle 5x + 30 = 10x + 50 \implies\)

\(\displaystyle 10x - 5x = 30 - 50 \implies 5x = - 20 \implies x = - 4.\)