Question

practice: add and subtract rational expressi
score: 1/3 · answered: 1/3
question 2
simplify completely.
\\(\frac{r + 1}{r + 4} + \frac{r - 6}{r + 7} = \\)
the domain is \\(r \
eq \\)
question help: video

Explanation:

Step1: Find a common denominator

The denominators are \( r + 4 \) and \( r + 7 \). The common denominator is \( (r + 4)(r + 7) \).
Rewrite each fraction with the common denominator:
\( \frac{(r + 1)(r + 7)}{(r + 4)(r + 7)} + \frac{(r - 6)(r + 4)}{(r + 4)(r + 7)} \)

Step2: Expand the numerators

Expand \( (r + 1)(r + 7) \): \( r^2 + 7r + r + 7 = r^2 + 8r + 7 \)
Expand \( (r - 6)(r + 4) \): \( r^2 + 4r - 6r - 24 = r^2 - 2r - 24 \)

Step3: Add the numerators

Add the expanded numerators: \( (r^2 + 8r + 7) + (r^2 - 2r - 24) = 2r^2 + 6r - 17 \)

Step4: Write the combined fraction

The combined fraction is \( \frac{2r^2 + 6r - 17}{(r + 4)(r + 7)} \) (this is simplified as much as possible since the numerator doesn't factor nicely with the denominator).

Step5: Find the domain restrictions

For the original rational expressions, the denominators cannot be zero. So \( r + 4
eq 0 \) implies \( r
eq -4 \), and \( r + 7
eq 0 \) implies \( r
eq -7 \).

Answer:

Simplified form: \( \frac{2r^2 + 6r - 17}{(r + 4)(r + 7)} \) (or expanded denominator \( \frac{2r^2 + 6r - 17}{r^2 + 11r + 28} \))
Domain restrictions: \( r
eq -4, -7 \) (for the domain fill - in, since it's a single box maybe expecting one? Wait, the problem has a single box for domain \( r
eq \), but actually there are two restrictions. But maybe a typo, but following the problem: if we consider the denominators, the values that make denominators zero are \( r = -4 \) and \( r = -7 \). But the box is single, maybe an error, but if we have to put one, but actually both. However, likely the problem expects both, but since it's a single box, maybe a mistake. But for the simplification:

Simplified expression: \( \boldsymbol{\frac{2r^2 + 6r - 17}{r^2 + 11r + 28}} \) (or factored denominator form)
Domain: \( r
eq \boldsymbol{-4, -7} \) (but if single box, maybe the problem expects both, but as per the input, the domain box is single, maybe a mistake, but the restrictions are \( r
eq -4 \) and \( r
eq -7 \))