The following rational equation has denominators that contain variables. For this equation, a. Write the value are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. \[ \frac{3}{x}+2=\frac{5}{4 x}+\frac{17}{8} \] 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.)

To find the restrictions on the variable \( x \) in the equation \[ \frac{3}{x} + 2 = \frac{5}{4x} + \frac{17}{8}, \] we need to identify the values of \( x \) that make any of the denominators zero. The denominators in the equation are \( x \) and \( 4x \). 1. For the denominator \( x \): - Setting \( x = 0 \) makes the denominator zero. 2. For the denominator \( 4x \): - Setting \( 4x = 0 \) also leads to \( x = 0 \). Thus, the only restriction on \( x \) is: \[ x = 0. \] So, the answer for part (a) is: \[ x = 0. \] Now, for part (b), we will solve the equation while keeping in mind that \( x \neq 0 \). Starting with the original equation: \[ \frac{3}{x} + 2 = \frac{5}{4x} + \frac{17}{8}. \] To eliminate the fractions, we can multiply through by the least common denominator (LCD), which is \( 8x \): \[ 8x \left( \frac{3}{x} \right) + 8x(2) = 8x \left( \frac{5}{4x} \right) + 8x \left( \frac{17}{8} \right). \] This simplifies to: \[ 24 + 16x = 10 + 17x. \] Now, we can rearrange the equation: \[ 24 - 10 = 17x - 16x, \] which simplifies to: \[ 14 = x. \] Thus, the solution to the equation, keeping in mind the restriction, is: \[ x = 14. \] In summary: - a. The restriction is \( x = 0 \). - b. The solution is \( x = 14 \).