Problem 16 - College Algebra

Question

Multiply or divide as indicated. $$\frac{6 x+9}{3 x-15} \cdot \frac{x-5}{4 x+6}$$

Step-by-Step Solution

Answer

After simplification, the final answer is 1.

Step 1: Simplify each fraction

The fractions can be simplified if they have common factors in the numerator and the denominator. Let's check for the first fraction, \( \frac{6x + 9} {3x - 15} \), it has common factor 3. We divided both the numerator and the denominator by 3, it becomes \( \frac{2x + 3} {x - 5} \). In the second fraction, \( \frac{x - 5} {4x + 6} \), has common factor 2. After dividing both the numerator and the denominator by 2, it becomes \( \frac{x - 5} {2x + 3} \)

Step 2: Multiply the fractions

Now, let's multiply the fractions. When multiplying fractions, we simply multiply the numerators together and the denominators together. Thus, \( \frac{2x + 3} {x - 5} * \frac{x - 5} {2x + 3} \) becomes \( \frac{(2x + 3)(x - 5)} {(x - 5)(2x + 3)} \)

Step 3: Simplify the final expression

The final expression, \( \frac{(2x + 3)(x - 5)} {(x - 5)(2x + 3)} \), can be simplified because it has the same expressions in the numerator as well as the denominator. After cancellation, our final answer is 1 since anything divided by itself is 1.