Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in $30$ minutes. While they took a break, Carl raked a third of the remaining lawn in $60$ minutes. Finally, Beth joined Carl and together they finished raking the lawn in $24$ minutes. If they each rake at a constant rate, how many hours would it have taken Alex to rake the entire lawn by himself?
1. Let \( a, b, c \) be the rates at which Alex, Beth, and Carl can rake the lawn in one hour, respectively. We need to find \( a \), the rate at which Alex can rake the lawn.

2. From the problem, we know that Alex and Beth together raked half of the lawn in 30 minutes. Therefore, their combined rate is:
   \[
   \frac{a + b}{2} = \frac{1}{2} \text{ (since they raked half the lawn in 30 minutes, which is half an hour)}
   \]
   Solving for \( a + b \):
   \[
   a + b = 1
   \]

3. Carl raked a third of the remaining lawn in 60 minutes. Since half the lawn was already raked, the remaining lawn is also half. Carl raked a third of this remaining half, which is:
   \[
   \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}
   \]
   Since Carl took 60 minutes (1 hour) to rake \(\frac{1}{6}\) of the lawn, his rate \( c \) is:
   \[
   c = \frac{1}{6}
   \]

4. Finally, Beth and Carl together finished raking the remaining lawn in 24 minutes. The remaining lawn after Carl raked is:
   \[
   \frac{1}{2} - \frac{1}{6} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
   \]
   Beth and Carl together raked this \(\frac{1}{3}\) of the lawn in 24 minutes, which is \(\frac{24}{60} = \frac{2}{5}\) hours. Therefore, their combined rate is:
   \[
   \frac{b + c}{\frac{2}{5}} = \frac{1}{3}
   \]
   Solving for \( b + c \):
   \[
   b + c = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}
   \]

5. We now have two equations:
   \[
   a + b = 1
   \]
   \[
   b + c = \frac{5}{6}
   \]
   Substituting \( c = \frac{1}{6} \) into the second equation:
   \[
   b + \frac{1}{6} = \frac{5}{6}
   \]
   Solving for \( b \):
   \[
   b = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}
   \]

6. Substituting \( b = \frac{2}{3} \) into the first equation:
   \[
   a + \frac{2}{3} = 1
   \]
   Solving for \( a \):
   \[
   a = 1 - \frac{2}{3} = \frac{1}{3}
   \]

7. Therefore, Alex's rate is \( \frac{1}{3} \) of the lawn per hour. To find how many hours it would take Alex to rake the entire lawn by himself:
   \[
   \text{Time} = \frac{1}{a} = \frac{1}{\frac{1}{3}} = 3 \text{ hours}
   \]

The final answer is \( \boxed{3} \) hours.