To solve this problem, let's carefully translate the given verbal expression into an algebraic expression step-by-step.
1. Identify the components:
- "Seven times a number": This is [tex]\( 7x \)[/tex] where [tex]\( x \)[/tex] is the number.
- "The sum of seven times a number and five": This is [tex]\( 7x + 5 \)[/tex].
2. Identify the components:
- "Negative two times the number": This is [tex]\( -2x \)[/tex].
- "The sum of negative two times the number and eleven": This is [tex]\( -2x + 11 \)[/tex].
3. Combine using division:
- The requirement is to divide "the sum of seven times a number and five" by "the sum of negative two times the number and eleven".
So the resulting algebraic expression is:
[tex]\[ \frac{7x + 5}{-2x + 11} \][/tex]
Now, let's match this expression with the given options:
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]
Clearly, the correct option is the fourth one: [tex]\(\frac{7x + 5}{-2x + 11}\)[/tex].
