To solve the rational equation 2/(x+2) = 9/8 - (5x)/(4x+8), we can begin by finding a common denominator on both sides of the equation, and then simplifying and rearranging terms to solve for x.
2/(x + 2) = 9/8 - (5x)/(4x + 8) (original equation)
16*2/(8(x+2)) = 2*9/8 - 5x/(4(x+2)) (multiply both sides by 8(x+2) to get a common denominator)
32/(x+2) = 9/4 - 5x/(4(x+2)) (simplify)
32/(x+2) = (9-5x)/(4(x+2)) (combine the fractions)
32 * 4(x+2) = (9-5x)(x+2) (cross-multiply)
128(x+2) = 9(x+2) - 5x(x+2) (distribute)
128x + 256 = 9x + 18 - 5x^2 - 10x (simplify and collect like terms)
5x^2 - 118x - 238 = 0 (rearrange to standard quadratic form)
We can then use the quadratic formula to solve for x:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Where a = 5, b = -118, and c = -238. Plugging in these values, we get:
x = (118 ± sqrt(118^2 - 4(5)(-238))) / (2*5)
x = (118 ± sqrt(14084)) / 10
x = (118 ± 118.6) / 10
So our two solutions for x are:
x = 23.72 or x = -9.52
We can check these solutions back in the original equation to confirm they work.