To determine which expression is equivalent to [tex]\(\frac{3x}{x+1}\)[/tex] divided by [tex]\(x+1\)[/tex], let's break down the process step by step.
1. Original Expression:
We start with [tex]\(\frac{3x}{x+1}\)[/tex] divided by [tex]\(x+1\)[/tex]. This can be written as:
[tex]\[
\frac{\frac{3x}{x+1}}{x+1}
\][/tex]
2. Simplify the Complex Fraction:
To simplify the complex fraction, we can multiply by the reciprocal. That is:
[tex]\[
\frac{\frac{3x}{x+1}}{x+1} = \frac{3x}{x+1} \cdot \frac{1}{x+1}
\][/tex]
This means our expression becomes:
[tex]\[
\frac{3x}{(x+1)^2}
\][/tex]
3. Compare with Given Choices:
We need to match this simplified form [tex]\(\frac{3x}{(x+1)^2}\)[/tex] with one of the given choices.
- Choice 1: [tex]\(\frac{3x}{x+1} \cdot \frac{1}{x+1}\)[/tex]
If we simplify this, we have:
[tex]\[
\frac{3x}{x+1} \cdot \frac{1}{x+1} = \frac{3x}{(x+1)^2}
\][/tex]
This matches our simplified form.
- Choice 2: [tex]\(\frac{3x}{x+1} \div \frac{1}{x+1}\)[/tex]
This involves division by a fraction, which is the same as multiplying by its reciprocal.
[tex]\[
\frac{3x}{x+1} \div \frac{1}{x+1} = \frac{3x}{x+1} \cdot \frac{x+1}{1} = 3x
\][/tex]
This does not match our simplified form.
- Choice 3: [tex]\(\frac{x+1}{1} \div \frac{3x}{x+1}\)[/tex]
Again, this involves division by a fraction:
[tex]\[
\frac{x+1}{1} \div \frac{3x}{x+1} = \frac{x+1}{1} \cdot \frac{x+1}{3x} = \frac{(x+1)^2}{3x}
\][/tex]
This does not match our simplified form.
- Choice 4: [tex]\(\frac{x+1}{3x} \cdot \frac{x+1}{1}\)[/tex]
Simplifying this:
[tex]\[
\frac{x+1}{3x} \cdot \frac{x+1}{1} = \frac{(x+1)^2}{3x}
\][/tex]
This does not match our simplified form.
4. Conclusion:
The correct equivalent expression is:
[tex]\[
\frac{3x}{x+1} \cdot \frac{1}{x+1}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{1}
\][/tex]
