\begin{tabular}{l} Solve \( \frac{1}{2}+\frac{1}{2 x}=\frac{x^{2}-7 x+10}{4 x} \) by rewriting the equation as a proportion. \\ Which proportion is equivalent to the original \\ equation? \\ \( \qquad \begin{array}{ll}\frac{x+2}{2 x}=\frac{x^{2}-7 x+10}{4 x} & \text { Name the true solution(s) to the equation. } \\ \frac{x-1}{2 x}=\frac{x^{2}-7 x+10}{4 x} & x=0 \\ x=1 \\ 2 x & \frac{x^{2}-7 x+10}{4 x} \\ x=8 \\ x=0, x=1, \text { and } x=8 \\ x=1 \text { and } x=8 \\ \text { There are no solutions. }\end{array} \) \\ \hline\end{tabular}

To solve the equation: \[ \frac{1}{2} + \frac{1}{2x} = \frac{x^2 - 7x + 10}{4x} \] **Step 1: Rewrite the Equation as a Proportion** First, express all terms with a common denominator to facilitate rewriting the equation as a proportion. \[ \frac{1}{2} = \frac{2x}{4x} \quad \text{and} \quad \frac{1}{2x} = \frac{2}{4x} \] Substituting these into the original equation: \[ \frac{2x}{4x} + \frac{2}{4x} = \frac{x^2 - 7x + 10}{4x} \] Combine the numerators on the left side: \[ \frac{2x + 2}{4x} = \frac{x^2 - 7x + 10}{4x} \] Since the denominators are the same and non-zero, set the numerators equal to each other: \[ 2x + 2 = x^2 - 7x + 10 \] **Step 2: Formulate the Equivalent Proportion** Rearrange the equation to standard quadratic form: \[ x^2 - 9x + 8 = 0 \] This can be written as the proportion: \[ \frac{x - 1}{2x} = \frac{x^2 - 7x + 10}{4x} \] **Step 3: Solve the Quadratic Equation** Factor the quadratic equation: \[ x^2 - 9x + 8 = (x - 1)(x - 8) = 0 \] Setting each factor to zero gives the potential solutions: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \] **Step 4: Verify Solutions** Ensure that the solutions do not make any denominators zero: - For \( x = 1 \): Denominators \( 2x \) and \( 4x \) become 2 and 4, respectively – valid. - For \( x = 8 \): Denominators \( 2x \) and \( 4x \) become 16 and 32, respectively – valid. **Conclusion** - **Equivalent Proportion:** \(\displaystyle \frac{x - 1}{2x} = \frac{x^2 - 7x + 10}{4x}\) - **True Solutions:** \( x = 1 \) and \( x = 8 \) **Final Answer:** After simplifying, an equivalent proportion is (x – 1)/2x = (x² – 7x + 10)/4x. The true solutions to the equation are x = 1 and x = 8.