Question

solve.
\\(\frac{1}{x^2 - 2x} - \frac{9}{x} = \frac{8}{x - 2}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\(\bigcirc\\) a. \\(x = \square\\)
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. the solution is all real numbers.
\\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Factor the denominator

Factor \(x^2 - 2x\) as \(x(x - 2)\). The equation becomes \(\frac{1}{x(x - 2)}-\frac{9}{x}=\frac{8}{x - 2}\).

Step2: Find the LCD

The least common denominator (LCD) of \(x(x - 2)\), \(x\), and \(x - 2\) is \(x(x - 2)\). Multiply each term by \(x(x - 2)\) to eliminate the fractions:

$$
LATEXBLOCK0
$$

Step3: Simplify and solve

Simplify the left - hand side: \(1-9x + 18=8x\). Combine like terms: \(19-9x = 8x\). Add \(9x\) to both sides: \(19=17x\). Then \(x=\frac{19}{17}\).

Step4: Check for extraneous solutions

We need to check if \(x = \frac{19}{17}\) makes the original denominators zero. For \(x^2-2x=x(x - 2)\), when \(x=\frac{19}{17}\), \(x
eq0\) and \(x - 2=\frac{19}{17}-2=\frac{19 - 34}{17}=-\frac{15}{17}
eq0\). So \(x=\frac{19}{17}\) is a valid solution.

Answer:

A. \(x=\frac{19}{17}\)