Final answer:
To solve the rational equation x/(2x+3) = (x+5)/(2x-3), multiply both sides by the common denominator, simplify, and then use the quadratic formula to find possible solutions. Check both solutions in the original equation to discard any that result in division by zero or are not reasonable.
Step-by-step explanation:
To solve the rational equation x/(2x+3) = (x+5)/(2x-3), first, we need to find a common denominator, which is (2x+3)(2x-3). We then multiply both sides of the equation by this common denominator to eliminate the fractions. This yields 2x(2x-3) = (2x+3)(x+5). Next, we expand both sides of the equation:
- 4x^2 - 6x = 2x^2 + 3x + 10x + 15
- 4x^2 - 6x = 2x^2 + 13x + 15
After arranging the terms, the equation becomes 2x^2 - 19x - 15 = 0. Factoring this quadratic yields potential solutions for x. However, one value of x may not satisfy the original equation (because it results in a division by zero or does not satisfy the original proportions), and so must be discarded, leaving one possible solution.
We can use the quadratic formula x = (-b ± √(b^2 - 4ac))/(2a) to solve for x. After determining the two potential x values, we need to substitute them back into the original equation to ensure that they do not result in division by zero.
If both values of x are valid, we accept them as solutions; otherwise, we discard the invalid one, leaving us with a single solution. Always remember to check the answer to see if it is reasonable and discard any that make no sense or are physically impossible.