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 \( \varnothing \).

Let's solve the rational equation step by step. ### Given Equation: \[ \frac{4}{x} = \frac{7}{4x} + 3 \] ### a. Finding Restrictions on \( x \): The denominators in the equation are \( x \) and \( 4x \). For the denominators to not be zero: \[ x \neq 0 \quad \text{and} \quad 4x \neq 0 \] Both conditions simplify to: \[ x \neq 0 \] So, the restriction is: \[ x = 0 \] **Answer for part a:** \[ x = 0 \] ### b. Solving the Equation: With the restriction \( x \neq 0 \), let's solve the equation: \[ \frac{4}{x} = \frac{7}{4x} + 3 \] **Step 1: Eliminate the denominators by multiplying both sides by \( 4x \):** \[ 4x \cdot \frac{4}{x} = 4x \cdot \frac{7}{4x} + 4x \cdot 3 \] Simplifying: \[ 16 = 7 + 12x \] **Step 2: Solve for \( x \):** \[ 16 - 7 = 12x \] \[ 9 = 12x \] \[ x = \frac{9}{12} = \frac{3}{4} \] **Final Solution:** The solution is \( x = \frac{3}{4} \), which satisfies the restriction \( x \neq 0 \). **Answer for part b:** **A.** The solution set is \( \left\{ \dfrac{3}{4} \right\} \).