Problem 16 - College Algebra Essentials

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

The answer is \( \frac{x + 3/2}{2x + 3} \)

Step 1: Simplify terms

The first task is to simplify the terms in each fraction as much as possible. In particular, the numerator and denominator of the first fraction, \( \frac{6x+9}{3x-15} \), can each be factored out by 3 to become \( \frac{2x+3}{x-5} \). Similarly, the numerator and denominator of the second fraction \( \frac{x-5}{4x+6} \) cannot be simplified

Step 2: Cancel common factors

Next, look for common factors in the numerators and denominators across both fractions and cancel them out. Here, we can see that \(x - 5\) is a common factor in the numerator of the second fraction and the denominator of the first fraction, so we can cancel these out. Our expression thus becomes \( \frac{2x + 3}{4x + 6} \)

Step 3: Simplify further

Again, simplify the terms as much as possible in the fraction. In this case, the numerator and the denominator have a common factor of 2. Therefore, the expression simplifies to \( \frac{x + 3/2}{2x + 3} \)