The following rationa equation has denom nators that contain variables. Keeping the restrictions in mind solve the equation \[ \frac{3}{x}+2=\frac{5}{4 x}+\frac{17}{3} \] a. What isiare the value or values of the variable that make(s) the denomin

To solve the equation \[ \frac{3}{x} + 2 = \frac{5}{4x} + \frac{17}{3} \] we first need to identify the restrictions on the variable \( x \) that make the denominators valid. The denominators in the equation are \( x \) and \( 4x \). ### Step 1: Identify Restrictions The denominators cannot be zero, so we set the following conditions: 1. \( x \neq 0 \) 2. \( 4x \neq 0 \) (which also implies \( x \neq 0 \)) Thus, the only restriction is \( x \neq 0 \). ### Step 2: Solve the Equation Next, we will solve the equation step by step. 1. **Eliminate the fractions**: To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is \( 12x \). \[ 12x \left( \frac{3}{x} + 2 \right) = 12x \left( \frac{5}{4x} + \frac{17}{3} \right) \] 2. **Distribute**: \[ 12x \cdot \frac{3}{x} + 12x \cdot 2 = 12x \cdot \frac{5}{4x} + 12x \cdot \frac{17}{3} \] This simplifies to: \[ 36 + 24x = 15 + 68x \] 3. **Rearrange the equation**: Move all terms involving \( x \) to one side and constant terms to the other side: \[ 36 - 15 = 68x - 24x \] This simplifies to: \[ 21 = 44x \] 4. **Solve for \( x \)**: \[ x = \frac{21}{44} \] ### Final Answer The value of the variable \( x \) that satisfies the equation is \[ x = \frac{21}{44} \] with the restriction that \( x \neq 0 \).