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The given expression is:

\[
\frac{x + 7}{9} \div \frac{4x + 28}{7}
\]

We can rewrite the division of fractions as multiplication by the reciprocal:

\[
\frac{x + 7}{9} \times \frac{7}{4x + 28}
\]

Now, simplify the expression:
- Factor \( 4x + 28 \) as \( 4(x + 7) \):

\[
\frac{x + 7}{9} \times \frac{7}{4(x + 7)}
\]

Next, notice that \( x + 7 \) appears in both the numerator and the denominator, so they cancel out:

\[
\frac{1}{9} \times \frac{7}{4} = \frac{7}{36}
\]

Thus, the simplified result is \( \frac{7}{36} \).

Now, based on the multiple-choice options:
- For option A, we have to exclude any values of \( x \) that would make the original denominators zero. Since \( x + 7 = 0 \) when \( x = -7 \), we exclude \( x = -7 \).
- For option B, it claims no numbers need to be excluded, which is incorrect because \( x = -7 \) must be excluded.

Thus, the correct answer is option A with \( \frac{7}{36} \) and \( x \neq -7 \).

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Do you want more details or have any questions? Here are some related questions for practice:

1. How do you solve fractions involving variables in the denominator?
2. What are the steps to simplify rational expressions?
3. How does factoring help in simplifying algebraic expressions?
4. What is the rule for dividing fractions?
5. What are common errors when canceling out terms in fractions?

**Tip**: Always check for values that make the denominator zero, as they need to be excluded from the solution!

Divide as indicated: (x + 7)/9 ÷ (4x + 28)/7.

Learn how to divide and simplify the algebraic expression (x + 7)/9 ÷ (4x + 28)/7 with a step-by-step solution. This lesson covers key concepts such as dividing fractions, factoring, and simplifying rational expressions. Suitable for students in Grades 8-10.

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