Rational Expresstons Sohting a work probiem using a rational equation \( 3 / 5 \) Taliyah Espanol Working together, two pumps can drain a certain pool in 6 hours. If it takes the older pump 14 hours to drain the pool by itself, how long will it take the newer pump to drain the pool on its own? Do not do any rounding. \( \square \) hours

Given that two pumps can drain a certain pool in 6 hours when working together, and the older pump can drain the pool by itself in 14 hours, we can set up a rational equation to solve for the time it takes the newer pump to drain the pool on its own. Let's denote the time it takes the newer pump to drain the pool on its own as \( x \) hours. The rate of work for the older pump is \( \frac{1}{14} \) of the pool per hour, and the rate of work for the newer pump is \( \frac{1}{x} \) of the pool per hour. When working together, the combined rate of work is \( \frac{1}{6} \) of the pool per hour. We can set up the following rational equation based on the rates of work: \[ \frac{1}{14} + \frac{1}{x} = \frac{1}{6} \] Now, we can solve this rational equation to find the value of \( x \), which represents the time it takes the newer pump to drain the pool on its own. Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{1}{14}+\frac{1}{x}=\frac{1}{6}\) - step1: Find the domain: \(\frac{1}{14}+\frac{1}{x}=\frac{1}{6},x\neq 0\) - step2: Move the constant to the right side: \(\frac{1}{x}=\frac{1}{6}-\frac{1}{14}\) - step3: Subtract the numbers: \(\frac{1}{x}=\frac{2}{21}\) - step4: Rewrite the expression: \(x=\frac{21}{2}\) - step5: Check if the solution is in the defined range: \(x=\frac{21}{2},x\neq 0\) - step6: Find the intersection: \(x=\frac{21}{2}\) The solution to the rational equation is \( x = \frac{21}{2} \) hours. Therefore, it will take the newer pump \( \frac{21}{2} \) hours to drain the pool on its own.