Questions: Find the least common denominator of x/(3x-6) and 2/(7x-14).

Transcript text: Find the least common denominator of $\frac{x}{3 x-6}$ and $\frac{2}{7 x-14}$.

Solution

Solution Steps

To find the least common denominator (LCD) of the given rational expressions, we need to factor the denominators and then determine the least common multiple (LCM) of these factors. The denominators are $3x - 6$ and $7x - 14$. We will factor each expression and then find the LCM of the factors.

Step 1: Factor the Denominators

We start with the denominators of the given rational expressions: \[ 3x - 6 \quad \text{and} \quad 7x - 14 \] Factoring these expressions, we have: \[ 3x - 6 = 3(x - 2) \quad \text{and} \quad 7x - 14 = 7(x - 2) \]

Step 2: Determine the Least Common Denominator

To find the least common denominator (LCD), we take the least common multiple (LCM) of the factored denominators: \[ \text{LCD} = \text{lcm}(3(x - 2), 7(x - 2)) \] The LCM of the coefficients $3$ and $7$ is $21$, and since both terms share the factor $(x - 2)$, the LCD is: \[ \text{LCD} = 21(x - 2) \]

Step 3: Expand the Least Common Denominator

Expanding the expression for the LCD gives: \[ \text{LCD} = 21x - 42 \]