`1/(2x+1)+5` Which of the following is equivalent to the expression above for x gt 0 ?

Options

A. `(2x+5)/(2x+1)`
B. `(2x+6)/(2x+1)`
C. `(10x+5)/(2x+1)`
D. `(10x+6)/(2x+1)`

Explanation

To solve the expression \( \frac{1}{2x+1} + 5 \) and find an equivalent expression for \( x > 0 \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ \frac{1}{2x+1} + 5 \]

### Step 2: Convert 5 into a fraction
To combine the terms, we can express 5 as a fraction with the same denominator:
\[ 5 = \frac{5(2x+1)}{2x+1} = \frac{10x + 5}{2x + 1} \]

### Step 3: Combine the fractions
Now we can add the two fractions:
\[ \frac{1}{2x+1} + \frac{10x + 5}{2x + 1} = \frac{1 + (10x + 5)}{2x + 1} \]

### Step 4: Simplify the numerator
Combine the terms in the numerator:
\[ 1 + 10x + 5 = 10x + 6 \]
So, we have:
\[ \frac{10x + 6}{2x + 1} \]

### Step 5: Final expression
Thus, the equivalent expression for \( \frac{1}{2x+1} + 5 \) is:
\[ \frac{10x + 6}{2x + 1} \]

### Conclusion
The final answer is \( \frac{10x + 6}{2x + 1} \), which corresponds to option D.

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