To find the least common denominator (LCD) of the fractions [tex]\(\frac{7}{3x} + \frac{2}{x} + \frac{x}{x-1}\)[/tex], we need to identify the denominators and determine the smallest common multiple.
Here are the denominators:
1. [tex]\(3x\)[/tex]
2. [tex]\(x\)[/tex]
3. [tex]\(x-1\)[/tex]
### Step-by-Step Solution:
1. Identify the Factors of Each Denominator:
- The first denominator [tex]\(3x\)[/tex] consists of factors 3 and [tex]\(x\)[/tex].
- The second denominator [tex]\(x\)[/tex] consists of the factor [tex]\(x\)[/tex].
- The third denominator [tex]\(x-1\)[/tex] is already a factored term.
2. Combine Factors:
- To account for all unique factors taking the highest power of each, we get:
- From [tex]\(3x\)[/tex], we include both 3 and [tex]\(x\)[/tex].
- From [tex]\(x\)[/tex], we have [tex]\(x\)[/tex] which is already included in [tex]\(3x\)[/tex].
- From [tex]\(x-1\)[/tex], we include [tex]\(x-1\)[/tex] as it is not a factor in the other denominators.
3. Construct the LCD:
- The least common multiple must include all the unique factors identified above.
- Therefore, the LCD is:
[tex]\[
\text{LCD} = 3 \cdot x \cdot (x - 1)
\][/tex]
4. Combine the Factors:
- Multiplying these gives the least common denominator:
[tex]\[
\text{LCD} = 3x(x - 1)
\][/tex]
Hence, the least common denominator for the fractions [tex]\(\frac{7}{3x} + \frac{2}{x} + \frac{x}{x-1}\)[/tex] is
[tex]\[
\boxed{3x(x-1)}
\][/tex]
