Let's break down the phrase "the sum of seven times a number and five, divided by the sum of negative two times the number and eleven" step by step using algebra.
1. Identify the variable:
- Let's denote the number as [tex]\( x \)[/tex].
2. Translate words into algebraic expressions:
- "The sum of seven times a number and five":
- Seven times a number is [tex]\( 7x \)[/tex].
- The sum of [tex]\( 7x \)[/tex] and five is [tex]\( 7x + 5 \)[/tex].
- "The sum of negative two times the number and eleven":
- Negative two times the number is [tex]\( -2x \)[/tex].
- The sum of [tex]\( -2x \)[/tex] and eleven is [tex]\( -2x + 11 \)[/tex].
3. Combine the two parts with the division operation:
- The phrase specifies that the first sum is divided by the second sum:
[tex]\[
\frac{7x + 5}{-2x + 11}
\][/tex]
Now that we have our algebraic expression, let's match it with the options provided:
1. [tex]\(\frac{-2x + 11}{7x + 5}\)[/tex]
2. [tex]\((-2x + 11) + \left(\frac{7x}{5}\right)\)[/tex]
3. [tex]\((7x + 5) + \left(\frac{-2x}{11}\right)\)[/tex]
4. [tex]\(\frac{7x + 5}{-2x + 11}\)[/tex]
The correct expression is [tex]\(\frac{7x + 5}{-2x + 11}\)[/tex], which matches option 4.
Therefore, the correct choice is:
[tex]\[
(7x + 5) \div (-2x + 11)
\][/tex]
This corresponds to option 4.
