\( \frac{7}{6 x}+\frac{x-5}{8 x^{2}} \) Select the correct choice below and fill in any answer boxes within your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) A. \( \frac{7}{6 x}+\frac{x-5}{8 x^{2}}=\frac{31 x-15}{7}, \frac{x-5}{8 x^{2}}=\square, \square \) B. no numbers must be excluded. B A

Simplify the expression by following steps: - step0: Solution: \(\frac{7}{6x}+\left(\frac{\left(x-5\right)}{8x^{2}}\right)\) - step1: Remove the parentheses: \(\frac{7}{6x}+\left(\frac{x-5}{8x^{2}}\right)\) - step2: Remove the parentheses: \(\frac{7}{6x}+\frac{x-5}{8x^{2}}\) - step3: Reduce fractions to a common denominator: \(\frac{7\times 4x}{6x\times 4x}+\frac{\left(x-5\right)\times 3}{8x^{2}\times 3}\) - step4: Multiply the numbers: \(\frac{7\times 4x}{24x\times x}+\frac{\left(x-5\right)\times 3}{8x^{2}\times 3}\) - step5: Multiply the numbers: \(\frac{7\times 4x}{24x\times x}+\frac{\left(x-5\right)\times 3}{24x^{2}}\) - step6: Multiply the terms: \(\frac{7\times 4x}{24x^{2}}+\frac{\left(x-5\right)\times 3}{24x^{2}}\) - step7: Transform the expression: \(\frac{7\times 4x+\left(x-5\right)\times 3}{24x^{2}}\) - step8: Multiply the terms: \(\frac{28x+\left(x-5\right)\times 3}{24x^{2}}\) - step9: Multiply the terms: \(\frac{28x+3x-15}{24x^{2}}\) - step10: Add the terms: \(\frac{31x-15}{24x^{2}}\) To simplify the expression \( \frac{7}{6 x}+\frac{x-5}{8 x^{2}} \), we follow these steps: 1. **Identify a common denominator**: The denominators are \( 6x \) and \( 8x^2 \). The least common denominator (LCD) is \( 24x^2 \). 2. **Rewrite each fraction with the common denominator**: - For \( \frac{7}{6x} \): \[ \frac{7}{6x} = \frac{7 \cdot 4x}{6x \cdot 4} = \frac{28x}{24x^2} \] - For \( \frac{x-5}{8x^2} \): \[ \frac{x-5}{8x^2} = \frac{(x-5) \cdot 3}{8x^2 \cdot 3} = \frac{3(x-5)}{24x^2} = \frac{3x - 15}{24x^2} \] 3. **Combine the fractions**: \[ \frac{28x}{24x^2} + \frac{3x - 15}{24x^2} = \frac{28x + 3x - 15}{24x^2} = \frac{31x - 15}{24x^2} \] Thus, the simplified expression is: \[ \frac{31x - 15}{24x^2} \] Now, let's fill in the answer boxes based on the choices provided: A. \( \frac{7}{6 x}+\frac{x-5}{8 x^{2}}=\frac{31 x-15}{24 x^{2}}, \frac{x-5}{8 x^{2}}=\frac{3x-15}{24x^{2}}, \frac{7}{6x}=\frac{28x}{24x^{2}} \) So the correct choice is A, and the filled answer boxes are: - \( \frac{x-5}{8 x^{2}}=\frac{3x-15}{24x^{2}} \) - \( \frac{7}{6x}=\frac{28x}{24x^{2}} \)