Questions: Solve the equation. x/(x-4) + 9 = 4/(x-4)

Solve the equation. x/(x-4) + 9 = 4/(x-4)
Transcript text: Solve the equation. \[ \frac{x}{x-4}+9=\frac{4}{x-4} \]
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Solution

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Solution Steps

Step 1: Eliminate the Denominator

To eliminate the fractions, we multiply both sides of the equation by \( x - 4 \): \[ (x - 4) \left( \frac{x}{x - 4} + 9 \right) = (x - 4) \left( \frac{4}{x - 4} \right) \] This simplifies to: \[ x + 9(x - 4) = 4 \]

Step 2: Simplify the Equation

Next, we expand and simplify the left side: \[ x + 9x - 36 = 4 \] Combining like terms gives: \[ 10x - 36 = 4 \]

Step 3: Solve for \( x \)

Now, we isolate \( x \) by adding 36 to both sides: \[ 10x = 40 \] Finally, we divide by 10: \[ x = 4 \]

Step 4: Check for Extraneous Solutions

Since the original equation has a denominator of \( x - 4 \), we must check if \( x = 4 \) is valid. Substituting \( x = 4 \) into the denominator results in division by zero, which is undefined. Therefore, \( x = 4 \) is not a valid solution.

Thus, the equation has no solutions.

Final Answer

\(\boxed{\text{No solutions}}\)

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