Question

solve.
\\(\frac{5}{x^2 - 4x} - \frac{2}{x} = \frac{9}{x - 4}\\)

Explanation:

Step1: Factor the denominator

Factor \(x^2 - 4x\) as \(x(x - 4)\). The equation becomes \(\frac{5}{x(x - 4)} - \frac{2}{x} = \frac{9}{x - 4}\).

Step2: Find the common denominator

The common denominator for the fractions is \(x(x - 4)\). Multiply each term by \(x(x - 4)\) to eliminate the denominators:
\(5 - 2(x - 4) = 9x\)

Step3: Simplify the left side

Expand and simplify: \(5 - 2x + 8 = 9x\), which simplifies to \(13 - 2x = 9x\).

Step4: Solve for x

Add \(2x\) to both sides: \(13 = 11x\). Then divide both sides by 11: \(x = \frac{13}{11}\).

Step5: Check for extraneous solutions

Check if \(x = \frac{13}{11}\) makes the original denominators zero. The denominators \(x^2 - 4x\), \(x\), and \(x - 4\) are not zero when \(x = \frac{13}{11}\), so it is a valid solution.

Answer:

\(x = \frac{13}{11}\)