Let's analyze the problem step by step.
We are given that Leo and Mary together can wash the car in 6 minutes. If Leo washes the car alone, it takes him 5 minutes longer than it takes Mary to wash the car alone. We represent the number of minutes it takes Mary to wash the car alone by [tex]\(x\)[/tex].
The given equation is:
[tex]\[
\frac{1}{x} + \frac{1}{x + 5} = \frac{1}{6}
\][/tex]
We need to find which value for [tex]\(x\)[/tex] must be discarded because it doesn't make sense in the given context. The options we are given are:
A. [tex]\(x = -10\)[/tex]
B. [tex]\(x = -5\)[/tex]
C. [tex]\(x = -3\)[/tex]
D. [tex]\(x = 0\)[/tex]
First, let's understand the context. The variable [tex]\(x\)[/tex] represents the time it takes Mary to wash the car alone. Time cannot be negative or zero because:
1. Negative time ([tex]\(x < 0\)[/tex]) does not exist in this context; it doesn't make sense for it to take a negative amount of time to complete a task.
2. Zero time ([tex]\(x = 0\)[/tex]) means that Mary washes the car instantly, which is also unrealistic.
Now, let’s evaluate each given option against these criteria:
- [tex]\(x = -10\)[/tex]: Negative, so it is not valid.
- [tex]\(x = -5\)[/tex]: Negative, so it is not valid.
- [tex]\(x = -3\)[/tex]: Negative, so it is not valid.
- [tex]\(x = 0\)[/tex]: Zero time, which is also not valid.
In conclusion, any value that is less than or equal to zero must be discarded in this context. Hence, the value that must be discarded from the rational equation because of the context is:
[tex]\[
\boxed{-10, -5, -3, 0}
\][/tex]
This includes all the options given (-10, -5, -3, and 0).
