Question

select the correct answer.
it would take charlene and gina 18 minutes to organize charlene’s comic book collection if they work together. if gina works alone, it would take her 15 minutes longer than it would take charlene working alone.
when x represents the number of minutes it takes charlene to organize the comic book collection working alone, the situation is represented by this equation:
\\(\frac{1}{x} + \frac{1}{x + 15} = \frac{1}{18}\\).
which value is a solution of the rational equation that must be discarded because of the context?
\\(\bigcirc\\) a. \\(x = 0\\)
\\(\bigcirc\\) b. \\(x = -9\\)
\\(\bigcirc\\) c. \\(x = -15\\)
\\(\bigcirc\\) d. \\(x = -30\\)

Explanation:

Step1: Analyze the context

In the context of time taken to complete a task, \( x \) represents the number of minutes Charlene takes alone. Time cannot be negative or zero (since taking 0 minutes or negative minutes to complete a task doesn't make sense in this real - world context).

Step2: Evaluate each option

• Option A: \( x = 0 \). If \( x=0 \), the term \( \frac{1}{x} \) is undefined (division by zero), so \( x = 0 \) is not a valid solution of the equation in terms of the equation's domain, but let's check other options.
• Option B: \( x=-9 \). If \( x = - 9 \), then the time Gina takes alone is \( x + 15=-9 + 15 = 6 \) minutes. But \( x=-9 \) represents a negative time for Charlene, which is not possible in the context of time taken to do a task.
• Option C: \( x=-15 \). If \( x=-15 \), then the term \( \frac{1}{x + 15}=\frac{1}{-15 + 15}=\frac{1}{0} \), which is undefined, so \( x=-15 \) is not in the domain of the original equation.
• Option D: \( x=-30 \). If \( x=-30 \), the time Gina takes alone is \( x + 15=-30 + 15=-15 \) minutes, which is negative. But we need to find the solution of the rational equation that must be discarded because of the context. First, we solve the equation \( \frac{1}{x}+\frac{1}{x + 15}=\frac{1}{18} \).

Multiply through by \( 18x(x + 15) \) (the least common denominator) to get:

\( 18(x + 15)+18x=x(x + 15) \)

Expand: \( 18x+270 + 18x=x^{2}+15x \)

Combine like terms: \( 36x+270=x^{2}+15x \)

Rearrange to standard quadratic form: \( x^{2}+15x-36x - 270 = 0\Rightarrow x^{2}-21x - 270=0 \)

Factor the quadratic: \( x^{2}-21x - 270=(x - 30)(x + 9)=0 \)

So the solutions are \( x = 30 \) and \( x=-9 \).

Now, in the context, \( x \) (time for Charlene alone) must be positive. \( x=-9 \) gives a negative time for Charlene (even though Gina's time would be \( - 9+15 = 6 \) minutes, Charlene's time can't be negative), so \( x=-9 \) is a solution of the rational equation but must be discarded because of the context.

Answer:

B. \( x = - 9 \)