- Define the expression as a fraction \(\frac{A}{B}\) where \(A = \frac{3x}{x+1}\) and \(B = x+1\).
- Apply the algebraic rule for division: \(\frac{A}{B} = A \cdot \frac{1}{B}\).
- Substitute the expressions for \(A\) and \(B\) to obtain \(\frac{3x}{x+1} \cdot \frac{1}{x+1}\).
- Select the matching option based on the derived result.
- The equivalent expression is \(\boxed{\frac{3x}{x+1} \cdot \frac{1}{x+1}}\).
Explanation
- Analysis of the Problem
We are asked to simplify the expression \(\frac{3x}{x+1}\) divided by \(x+1\). Let's denote the first part as \(A = \frac{3x}{x+1}\) and the divisor as \(B = x+1\). The entire problem is to compute the expression \(\frac{A}{B}\).
- Applying the Division Rule
Dividing any value by a number or expression \(B\) is equivalent to multiplying that value by the reciprocal of \(B\), which is \(\frac{1}{B}\). Applying this rule to our problem:
\[\frac{A}{B} = A \cdot \frac{1}{B} = \frac{3x}{x+1} \cdot \frac{1}{x+1}\]
This follows directly from the definition of division in algebra, which states that for any non-zero \(B\), \(\frac{A}{B} = A \times \frac{1}{B}\).
- Verification with Given Options
Now, let's compare our result, \(\frac{3x}{x+1} \cdot \frac{1}{x+1}\), with the provided options:
- \(\frac{3x}{x+1} \cdot \frac{1}{x+1}\)
- \(\frac{3x}{x+1} + \frac{1}{x+1}\)
- \(\frac{x+1}{1} + \frac{3x}{x+1}\)
- \(\frac{x+1}{3x} \cdot \frac{x+1}{1}\)
The first option matches our derived expression perfectly. Thus, the expression \(\frac{3x}{x+1}\) divided by \(x+1\) is indeed \(\frac{3x}{x+1} \cdot \frac{1}{x+1}\).
Examples
Understanding how to divide expressions is like splitting a bulk order. If you have \(\frac{3x}{x+1}\) pounds of sugar that needs to be distributed into \(x+1\) bags, you calculate the amount per bag by dividing the total weight by the number of bags: \(\frac{\text{Total}}{\text{Number of bags}} = \frac{3x}{x+1} \div (x+1)\). By converting this division into multiplication by the reciprocal, you get the weight per bag as \(\frac{3x}{x+1} \cdot \frac{1}{x+1}\), helping you efficiently determine how to package your goods.