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

To find the least common denominator (LCD) of the fractions \( \frac{4 x}{x-3} \) and \( \frac{x}{x-5} \), we look at the denominators \( x-3 \) and \( x-5 \). The LCD is simply the product of these two factors, since they are both linear and do not share any common factors. Therefore, the least common denominator is \( (x-3)(x-5) \). Additionally, just to emphasize, the LCD is crucial when adding or subtracting fractions, as it allows us to have a common base to work from. By multiplying each fraction by the appropriate factor, we can rewrite them with the same denominator, making combining them a breeze! When working with denominators, a common mistake is forgetting to factor completely or to overlook a simpler form of the LCD. Always be on the lookout for possible simplifications that could save you time and effort!