Answer:
1. The error occurred in step 2. (See below for the full explanation).
2. The valid solution is x = 0. (See below for the full explanation).
Step-by-step explanation:
The error occurred during the multiplication in step 2, where only the fractional term on the left side was multiplied by (x - 8), instead of all the terms, leading to the rest of the calculation and the final result being incorrect.
When applying the Multiplication Property of Equality to an algebraic equation, we need to ensure that all terms on both sides are multiplied by the same non-zero algebraic expression to maintain equality. So, in this case, the property was not applied correctly.
The correct solution to the problem is as follows:
[tex]\begin{aligned}-\dfrac{8}{x-8} - 1 &= \dfrac{x}{x-8}\\\\-\dfrac{8}{x-8}(x-8) - 1(x-8) &= \dfrac{x}{x-8}(x-8)\\\\-8-x+8&=x\\\\-x+x&=x+x\\\\0&=2x\\\\\dfrac{0}{2}&=\dfrac{2x}{2}\\\\0&=x\end{aligned}[/tex]
In step 2, we multiplied all terms by (x - 8), which cancelled the fractions.
In step 4, we added x to both sides of the equation.
In step 6, we divided both sides of the equation by 2.
An extraneous solution is a solution that is produced by solving the problem, but is not a valid solution to the problem. If we substitute x = 0 into the original equation we get:
[tex]\begin{aligned}-\dfrac{8}{0-8} - 1 &= \dfrac{0}{0-8}\\\\-\dfrac{8}{-8} - 1 &=0\\\\1-1&=0\\\\0&=0\end{aligned}[/tex]
So, x = 0 is a valid solution, and there are no extraneous solutions.
