Questions: Solve the following equation. 1/x - x/81 = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.

$Solve the following equation. 1/x - x/81 = 0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.$

Transcript text: Solve the following equation. \[ \frac{1}{x}-\frac{x}{81}=0 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \}. (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is $\varnothing$.

Solution

Solution Steps

To solve the equation $\frac{1}{x} - \frac{x}{81} = 0$, we first find a common denominator and combine the terms. Then, we set the resulting expression equal to zero and solve for $x$.

Step 1: Set Up the Equation

We start with the equation

\[ \frac{1}{x} - \frac{x}{81} = 0. \]

Step 2: Combine Terms

To solve for $x$, we can rearrange the equation:

\[ \frac{1}{x} = \frac{x}{81}. \]

Step 3: Cross-Multiply

Cross-multiplying gives us:

\[ 81 = x^2. \]

Step 4: Solve for $x$

Taking the square root of both sides, we find:

\[ x = \pm 9. \]