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

Solve the equation and check your solution. 1/x + 4/(x-8) = 0
Transcript text: Solve the equation and check your solution. \[ \frac{1}{x}+\frac{4}{x-8}=0 \]
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Solution

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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.

Step 1: Find a Common Denominator

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:

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

Step 2: Combine the Fractions

Combine the fractions over the common denominator:

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

Step 3: Solve the Numerator

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

\[ 5x - 8 = 0 \]

Solving this equation gives:

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

Step 4: Check for Validity

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

The solution to the equation is \(\boxed{x = \frac{8}{5}}\).

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