Let's carefully analyze the problem step-by-step:
1. Lina travels for the first 80 miles at a speed of [tex]\( s \)[/tex] mph. Therefore, the time for this part of the trip can be calculated as [tex]\( \frac{80}{s} \)[/tex] hours.
2. For the remaining 50 miles, she travels 10 mph faster than her initial speed. Hence, her speed for this part is [tex]\( s + 10 \)[/tex] mph. The time for this part of the trip can be calculated as [tex]\( \frac{50}{s+10} \)[/tex] hours.
To find the total time for her trip, we simply add the time for the first part and the time for the second part:
[tex]\[
\text{Total time} = \frac{80}{s} + \frac{50}{s+10}
\][/tex]
Now, let's evaluate each given expression to see which ones match this correct expression.
1. [tex]\(\frac{130 s-800}{s(s-10)}\)[/tex]:
- This expression does not match our derived formula. It's not structured correctly as a sum of fractions but rather a single complex fraction, so it is incorrect.
2. [tex]\(\frac{80}{x+10}+\frac{50}{x}\)[/tex]:
- This expression incorrectly swaps [tex]\(s\)[/tex] and [tex]\(s+10\)[/tex] in the denominators as compared to our derived formula. Hence, it is incorrect.
3. [tex]\(\frac{130 x+800}{a(a+10)}\)[/tex]:
- This expression is not presented in the form of the sum of the time durations as calculated. Therefore, it is incorrect.
4. [tex]\(\frac{130}{a(x+10)}\)[/tex]:
- This expression has mismatched variables and incorrect structure compared to our calculated formula. Hence, it is incorrect.
5. [tex]\(\frac{80}{x}+\frac{50}{x+10}\)[/tex]:
- This expression correctly represents the total time of the trip as [tex]\(\frac{80}{s}\)[/tex] for the first part and [tex]\(\frac{50}{s+10}\)[/tex] for the second part. Hence, it is correct.
Therefore, the correct expression that represents the total time of Lina's trip is:
[tex]\[
\frac{80}{x}+\frac{50}{x+10}
\][/tex]
which corresponds to:
[tex]\[
\boxed{\frac{80}{x}+\frac{50}{x+10}}
\][/tex]
