Given \(\frac{14x+22}{x+2}=-3x+1\), compute the sum of all the possible values of \(x\). Express your answer as a fraction.
Solve for x:
(14 x + 22)/(x + 2) = 1 - 3 x
Multiply both sides by x + 2:
14 x + 22 = (1 - 3 x) (x + 2)
Expand out terms of the right hand side:
14 x + 22 = -3 x^2 - 5 x + 2
Subtract -3 x^2 - 5 x + 2 from both sides:
3 x^2 + 19 x + 20 = 0
The left hand side factors into a product with two terms:
(x + 5) (3 x + 4) = 0
Split into two equations:
x + 5 = 0 or 3 x + 4 = 0
Subtract 5 from both sides:
x = -5 or 3 x + 4 = 0
Subtract 4 from both sides:
x = -5 or 3 x = -4
Divide both sides by 3:
Answer: | x = -5 or x = -4/3
