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: $$ \frac{1}{z}+\frac{1}{x+15}=\frac{1}{18} $$. Which value is a solution of the rational equation that must be discarded because of the context? A. $$ x=0 $$ B. $$ x=-9 $$ C. $$ x=-15 $$ D. $$ x=-30 $$

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UpStudy AI · Accepted Answer

To solve the problem, let's first interpret the given equation correctly. The situation can be modeled by the equation:

$$
\frac{1}{x} + \frac{1}{x + 15} = \frac{1}{18}
$$

Here, $$ x $$ represents the time it takes Charlene to organize the comic book collection alone, and $$ x + 15 $$ represents the time it takes Gina to do the same.

**Solving the Equation:**

1. **Multiply both sides by $$ 18x(x + 15) $$ to eliminate the denominators:**

$$
18(x + 15) + 18x = x(x + 15)
$$

2. **Simplify the equation:**

$$
18x + 270 + 18x = x^2 + 15x
$$

$$
36x + 270 = x^2 + 15x
$$

3. **Rearrange to form a standard quadratic equation:**

$$
x^2 - 21x - 270 = 0
$$

4. **Solve the quadratic equation using the quadratic formula:**

$$
x = \frac{21 \pm \sqrt{441 + 1080}}{2} = \frac{21 \pm 39}{2}
$$

This yields two solutions:

$$
x = 30 \quad \text{and} \quad x = -9
$$

**Contextual Interpretation:**

- **$$ x = 30 $$ minutes:** This is a valid solution because time cannot be negative.
  
- **$$ x = -9 $$ minutes:** This is **not valid** because time cannot be negative.

Among the provided options:

- **A. $$ x = 0 $$**
- **B. $$ x = -9 $$**
- **C. $$ x = -15 $$**
- **D. $$ x = -30 $$**

Only **$$ x = -9 $$** is a solution to the equation that must be discarded due to the context of the problem.

**Answer:**  
**B. $$ x = -9 $$**
$$ x = -9 $$ must be discarded because it represents a negative time, which is not possible in this context.

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