To solve the problem, we need to evaluate the expression given by the formula: \[ m * n = \left( \frac{m}{n} - \frac{n}{m} \right) \] where \( m \) and \( n \) are real numbers. In this case, we are asked to evaluate \( -3 * 4 \). ### Step 1: Identify the values of \( m \) and \( n \) Here, we have: - \( m = -3 \) - \( n = 4 \) ### Step 2: Substitute the values into the formula Now, we substitute \( m \) and \( n \) into the formula: \[ -3 * 4 = \left( \frac{-3}{4} - \frac{4}{-3} \right) \] ### Step 3: Calculate each fraction 1. Calculate \( \frac{-3}{4} \): \[ \frac{-3}{4} = -0.75 \] 2. Calculate \( \frac{4}{-3} \): \[ \frac{4}{-3} = -\frac{4}{3} \approx -1.3333 \] ### Step 4: Combine the fractions Now we need to combine these two fractions: \[ -3 * 4 = \left( -\frac{3}{4} - \left(-\frac{4}{3}\right) \right) = -\frac{3}{4} + \frac{4}{3} \] ### Step 5: Find a common denominator To add these fractions, we need a common denominator. The least common multiple of 4 and 3 is 12. - Convert \( -\frac{3}{4} \) to a fraction with a denominator of 12: \[ -\frac{3}{4} = -\frac{3 \times 3}{4 \times 3} = -\frac{9}{12} \] - Convert \( \frac{4}{3} \) to a fraction with a denominator of 12: \[ \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12} \] ### Step 6: Add the fractions Now we can add the two fractions: \[ -\frac{9}{12} + \frac{16}{12} = \frac{16 - 9}{12} = \frac{7}{12} \] ### Final Answer Thus, the evaluation of \( -3 * 4 \) is: \[ -3 * 4 = \frac{7}{12} \] ### Conclusion The correct option is **C. \( \frac{7}{12} \)**. ### Explanation of Other Options - **Option A: \( -\frac{25}{12} \)**: This option is incorrect because it does not reflect the correct calculation of the fractions involved. - **Option B: \( -\frac{7}{12} \)**: This option is incorrect as it has the wrong sign; our calculation resulted in a positive value. - **Option D: \( \frac{25}{12} \)**: This option is also incorrect as it does not match our calculated result. ### Revision Summary - The operation defined is \( m * n = \left( \frac{m}{n} - \frac{n}{m} \right) \). - Substitute the values of \( m \) and \( n \) into the formula. - Convert fractions to a common denominator to perform addition. - The final result for \( -3 * 4 \) is \( \frac{7}{12} \).