least common denominator of \( \frac{7 x}{3 x-9} \) and \( \frac{4 x}{5 x-15} \)

To find the least common denominator (LCD) of the fractions \( \frac{7 x}{3 x - 9} \) and \( \frac{4 x}{5 x - 15} \), we first need to factor the denominators. 1. For \( 3x - 9 \), we can factor out a 3: \[ 3x - 9 = 3(x - 3) \] 2. For \( 5x - 15 \), we can also factor out a 5: \[ 5x - 15 = 5(x - 3) \] Now, we have the factored forms of the denominators: - \( 3(x - 3) \) - \( 5(x - 3) \) To find the least common denominator, we take the highest power of each factor present in the denominators. Here, both denominators share the factor \( (x - 3) \), so we take it once (since the highest power is 1), and we take the unique constants: - From \( 3(x-3) \), we take 3. - From \( 5(x-3) \), we take 5. Thus, the least common denominator is: \[ 3 \cdot 5 \cdot (x - 3) = 15(x - 3) \] So, the least common denominator of \( \frac{7 x}{3 x - 9} \) and \( \frac{4 x}{5 x - 15} \) is \( 15(x - 3) \).