Question

Subtract the following rational expressions. (1 point) (x+1)/x+3 - 5/2x+6 = □ /2x+6

Answer 1

To subtract the given rational expressions \(\frac{x+1}{x+3} - \frac{5}{2x+6}\), we first need to find a common denominator.

The expression \(2x + 6\) can be factored as:

\[
2x + 6 = 2(x + 3)
\]

So, the common denominator between \(x + 3\) and \(2(x + 3)\) is \(2(x + 3)\).

Now, we can rewrite both fractions with the common denominator:

1. The first fraction \(\frac{x + 1}{x + 3}\) can be rewritten as:

\[
\frac{x + 1}{x + 3} = \frac{(x + 1) \cdot 2}{(x + 3) \cdot 2} = \frac{2(x + 1)}{2(x + 3)}
\]

2. The second fraction is already in the new common denominator:

\[
\frac{5}{2x + 6} = \frac{5}{2(x + 3)}
\]

Now we can perform the subtraction:

\[
\frac{2(x + 1)}{2(x + 3)} - \frac{5}{2(x + 3)} = \frac{2(x + 1) - 5}{2(x + 3)}
\]

Next, we simplify the numerator:

\[
2(x + 1) - 5 = 2x + 2 - 5 = 2x - 3
\]

Therefore, we have:

\[
\frac{2(x + 1) - 5}{2(x + 3)} = \frac{2x - 3}{2(x + 3)}
\]

Thus, the subtraction results in:

\[
\frac{2x - 3}{2x + 6}
\]

So, the final answer is:

\[
\frac{2x - 3}{2x + 6}
\]

This means that the box \(□\) can be filled with \(2x - 3\):

\[
\boxed{2x - 3}
\]