To paint a wall, Dennis takes \( x \) days to complete one third of the wall when Dennis works alone while Carl takes one more day to complete half of the job when Carl works alone. If it takes 3 days for Dennis and Carl to complete the whole wall when they work together, find \( x \).

To solve this problem, we first need to determine the work rates of Dennis and Carl. 1. **Work Rate of Dennis:** Since Dennis takes \( x \) days to complete one-third of the wall, he can complete the whole wall in \( 3x \) days. Thus, his rate of work is: \[ \text{Dennis's rate} = \frac{1 \text{ wall}}{3x \text{ days}} = \frac{1}{3x} \text{ walls per day} \] 2. **Work Rate of Carl:** Carl takes one more day than Dennis to complete half of the wall. Therefore, if Dennis takes \( 3x \) days for one wall, then Carl takes \( 3x + 1 \) days to complete half of it. Thus, the time he takes to complete the entire wall is: \[ \text{Carl's time for 1 wall} = 2(3x + 1) = 6x + 2 \text{ days} \] His rate of work is then: \[ \text{Carl's rate} = \frac{1}{6x + 2} \text{ walls per day} \] 3. **Combined Work Rate:** When Dennis and Carl work together for 3 days, they complete the wall, which means: \[ 3 \left( \frac{1}{3x} + \frac{1}{6x + 2} \right) = 1 \] 4. **Setting Up the Equation:** Using the combined work rate: \[ \frac{3}{3x} + \frac{3}{6x + 2} = 1 \] We can simplify: \[ \frac{1}{x} + \frac{3}{6x + 2} = 1 \] 5. **Finding a Common Denominator:** The common denominator will be \( x(6x + 2) \). Multiplying throughout by this common denominator gives: \[ (6x + 2) + 3x = x(6x + 2) \] Simplifying the left side results in: \[ 9x + 2 = 6x^2 + 2x \] 6. **Rearranging to Form a Quadratic Equation:** Rearranging gives: \[ 6x^2 - 7x - 2 = 0 \] 7. **Using the Quadratic Formula:** We will now apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6 \), \( b = -7 \), and \( c = -2 \): \[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 6 \cdot (-2)}}{2 \cdot 6} \] \[ x = \frac{7 \pm \sqrt{49 + 48}}{12} \] \[ x = \frac{7 \pm \sqrt{97}}{12} \] 8. **Finding the Positive Solution:** Since \( x \) represents days, we are only interested in the positive root: \[ x = \frac{7 + \sqrt{97}}{12} \] Thus, the value of \( x \) is approximately \( 1.72 \) days when calculated numerically.