The following rational equation has denominators that contain variables. For this equation a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. \( \frac{4}{x}=\frac{7}{4 x}+3 \) a. What is/are the value or values of the variable that make(s) the denominators zero? \( x=\square \) (Simplify your answer. Use a comma to separate answers as needed.) b. Solve the equation. 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 \( \{x \mid x \) is a real number \( \} \). C. The solution set is \( \varnothing \).

To find the values of the variable that make the denominators zero, we set each denominator equal to zero. For \(\frac{4}{x}\), the denominator is \(x\), so setting \(x = 0\) will make this expression undefined. Also, for \(\frac{7}{4x}\), the denominator \(4x\) also turns zero when \(x = 0\). Therefore, the restriction on the variable is \(x = 0\). Now, moving on to solve the equation \(\frac{4}{x} = \frac{7}{4x} + 3\). First, we will eliminate the fractions by multiplying every term by \(4x\) (keeping in mind that \(x \neq 0\)): \[ 4 \cdot 4 = 7 + 12x \] This simplifies to \(16 = 7 + 12x\). By rearranging the equation, we have: \[ 12x = 16 - 7 \] \[ 12x = 9 \] Solving for \(x\): \[ x = \frac{9}{12} = \frac{3}{4} \] Thus, for part b, the solution set is \( \{ \frac{3}{4} \} \). So the final answers are: a. \( x = 0 \) b. A. The solution set is \( \{\frac{3}{4}\} \).