Simplify each complex fraction. See Examples 5 and 6. [ (−4/3) + 12/5 ] / [ 1 − (−4/3)(12/5) ]

Verified step by step guidance

1. Identify the complex fraction structure: the numerator is \(\left(-\frac{4}{3}\right) + \frac{12}{5}\) and the denominator is \(1 - \left(-\frac{4}{3}\right) \times \frac{12}{5}\).
2. Simplify the numerator by finding a common denominator and adding the fractions: calculate \(-\frac{4}{3} + \frac{12}{5}\).
3. Simplify the denominator by first multiplying the fractions inside the parentheses: compute \(\left(-\frac{4}{3}\right) \times \frac{12}{5}\), then subtract this product from 1.
4. Rewrite the complex fraction as a division of two simplified fractions: \(\frac{\text{numerator}}{\text{denominator}}\) becomes \(\text{numerator} \div \text{denominator}\).
5. Perform the division by multiplying the numerator by the reciprocal of the denominator, then simplify the resulting fraction by reducing common factors.

Key Concepts

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression to eliminate the smaller fractions by finding common denominators or multiplying numerator and denominator appropriately.

Operations with Fractions

Adding, subtracting, multiplying, and dividing fractions require finding common denominators or multiplying numerators and denominators directly. Understanding how to manipulate fractions is essential to simplify expressions accurately.

Order of Operations and Negative Signs

Applying the correct order of operations (PEMDAS) ensures accurate simplification, especially when dealing with negative signs and parentheses. Careful handling of negative fractions and multiplication prevents sign errors in the final result.