Final answer:
The rational equation (4)/(x+9) = (2)/(x-5) is solved by finding a common denominator, cross-multiplying to eliminate fractions, and then simplifying to solve for x. The solution to the equation after simplification is x = 19.
Step-by-step explanation:
To solve the rational equation (4)/(x+9) = (2)/(x-5), we need to find a common denominator and then cross-multiply to get a quadratic equation. After setting the equation equal to zero, we would typically use the quadratic formula to find the values of x that satisfy the equation. However, we can quickly see that this equation can be simplified by multiplying both sides by the product of the denominators, which leads to two linear terms.
First, multiply both sides by (x+9)(x-5) to eliminate the fractions:
4(x-5) = 2(x+9)
Now, we distribute the numbers across the parentheses:
Next, we bring all terms containing x to one side and constants to the other side:
Finally, we divide both sides by 2 to solve for x:
Thus, the solution to the equation is x = 19. We must also ensure that this solution does not make any denominators in the original equation equal to zero. Since 19 + 9 and 19 - 5 are non-zero, x = 19 is a valid solution.