To determine which phrase represents the algebraic expression [tex]\(\frac{3 \rho + 6}{7 \rho - 9}\)[/tex], we'll analyze each phrase provided and compare it with the given expression.
1. Phrase 1: "The sum of six times a number and three, multiplied by the difference of nine times the number and seven."
Algebraic expression: [tex]\((6 \rho + 3) \cdot (9 \rho - 7)\)[/tex]
This expression involves multiplication, not division.
2. Phrase 2: "The sum of six times a number and three, divided by the difference of nine times the number and seven."
Algebraic expression: [tex]\(\frac{6 \rho + 3}{9 \rho - 7}\)[/tex]
Though this involves division, the coefficients do not match those in the given expression.
3. Phrase 3: "The sum of three times a number and six, multiplied by the difference of seven times the number and nine."
Algebraic expression: [tex]\((3 \rho + 6) \cdot (7 \rho - 9)\)[/tex]
This expression involves multiplication, not division.
4. Phrase 4: "The sum of three times a number and six, divided by the difference of seven times the number and nine."
Algebraic expression: [tex]\(\frac{3 \rho + 6}{7 \rho - 9}\)[/tex]
This matches the given algebraic expression exactly.
Based on this detailed analysis, the correct phrase that represents the algebraic expression [tex]\(\frac{3 \rho + 6}{7 \rho - 9}\)[/tex] is:
Phrase 4: "The sum of three times a number and six, divided by the difference of seven times the number and nine."
