Questions: Solve the equation and check your solution. 1/x + 4/(x-8) = 0

Solution Steps

To solve the equation \(\frac{1}{x} + \frac{4}{x-8} = 0\), we first find a common denominator, which is \(x(x-8)\). We then combine the fractions and set the numerator equal to zero to solve for \(x\). Finally, we check the solution to ensure it does not make any denominator zero.

To solve the equation \(\frac{1}{x} + \frac{4}{x-8} = 0\), we first find a common denominator, which is \(x(x-8)\). This allows us to combine the fractions:

\[ \frac{x-8}{x(x-8)} + \frac{4x}{x(x-8)} = 0 \]

Combine the fractions over the common denominator:

\[ \frac{x-8 + 4x}{x(x-8)} = \frac{5x - 8}{x(x-8)} = 0 \]

Set the numerator equal to zero and solve for \(x\):

\[ 5x - 8 = 0 \]

Solving this equation gives:

\[ 5x = 8 \quad \Rightarrow \quad x = \frac{8}{5} \]

Check that the solution does not make any denominator zero. The denominators are \(x\) and \(x-8\). For \(x = \frac{8}{5}\), neither \(x\) nor \(x-8\) is zero, so the solution is valid.

Final Answer