Question

Solve the rational equation \( \frac{1}{x-4} + \frac{x}{x-2} = \frac{2}{(x-2)(x-4)} \) using the least common denominator (LCD).

a) x = 3
b) x = 5
c) x = 6
d) x = 8

Answer 1

The solution to the rational equation is x = 5.

Step-by-step explanation:

To solve the given rational equation
`\( (1)/(x-4) + (x)/(x-2) = (2)/((x-2)(x-4)) \)` using the least common denominator (LCD), first, identify the LCD, which is (x-2)(x-4) since it includes the factors of both denominators. Next, multiply both sides of the equation by the LCD to clear the fractions. This results in the equation becoming a polynomial equation.

Now, simplify the equation and combine like terms. After simplifying, the equation reduces to a linear equation. Solve for x by isolating it on one side of the equation. In this case, x = 5 is the solution.

The solution x = 5 is obtained by ensuring that the equation is satisfied when x is substituted back into the original rational equation. It is crucial to check for extraneous solutions and ensure that the denominator is not equal to zero, which could lead to undefined values. Therefore, x = 5 is the valid solution to the given rational equation, satisfying the original equation and avoiding any division by zero.