To determine which phrase represents the algebraic expression [tex]\(\frac{3p + 6}{7p - 9}\)[/tex], we need to carefully analyze each phrase and match it to the given algebraic expression.
Let's break down each part of the expression step-by-step.
### Given Expression
[tex]\[
\frac{3p + 6}{7p - 9}
\][/tex]
#### Numerator: [tex]\(3p + 6\)[/tex]
- This part is the sum of three times a number [tex]\(p\)[/tex] and six.
#### Denominator: [tex]\(7p - 9\)[/tex]
- This part is the difference of seven times a number [tex]\(p\)[/tex] and nine.
Now, let's review the given phrases:
1. "the sum of six times a number and three, multiplied by the difference of nine times the number and seven"
- Numerator: [tex]\(6p + 3\)[/tex]
- Denominator: [tex]\(9p - 7\)[/tex]
- This does not match [tex]\(\frac{3p + 6}{7p - 9}\)[/tex].
2. "the sum of six times a number and three, divided by the difference of nine times the number and seven"
- Numerator: [tex]\(6p + 3\)[/tex]
- Denominator: [tex]\(9p - 7\)[/tex]
- This does not match [tex]\(\frac{3p + 6}{7p - 9}\)[/tex].
3. "the sum of three times a number and six, multiplied by the difference of seven times the number and nine"
- Numerator: [tex]\(3p + 6\)[/tex]
- Denominator: [tex]\(7p - 9\)[/tex] (but indicates multiplication instead of division)
- This is not correct because the phrase indicates multiplication rather than division.
4. "the sum of three times a number and six, divided by the difference of seven times the number and nine"
- Numerator: [tex]\(3p + 6\)[/tex]
- Denominator: [tex]\(7p - 9\)[/tex]
- This matches exactly with [tex]\(\frac{3p + 6}{7p - 9}\)[/tex] since it correctly identifies the division operation.
### Conclusion
The phrase that correctly represents the given algebraic expression [tex]\(\frac{3p + 6}{7p - 9}\)[/tex] is:
[tex]\[
\boxed{\text{the sum of three times a number and six, divided by the difference of seven times the number and nine}}
\][/tex]
