To solve this problem, we can use the concept of rates to determine how long it will take Jackie and Lisa to paint the living room together.

1. Jackie's rate: Jackie can paint the room in 5 hours. Her rate of painting is: $\text{Jackie's rate} = \frac{1 \text{ room}}{5 \text{ hours}} = \frac{1}{5}$

2. Lisa's rate: Lisa can paint the room in 6 hours. Her rate of painting is: $\text{Lisa's rate} = \frac{1 \text{ room}}{6 \text{ hours}} = \frac{1}{6}$

When they work together, their combined rate is the sum of their individual rates. So, the combined rate is: $\text{Combined rate} = \frac{1}{5} + \frac{1}{6}$

To add these fractions, we need a common denominator. The least common denominator of 5 and 6 is 30. We rewrite each fraction: $\frac{1}{5} = \frac{6}{30}, \quad \frac{1}{6} = \frac{5}{30}$

Now, add them together: $\text{Combined rate} = \frac{6}{30} + \frac{5}{30} = \frac{11}{30}$

This means that together, Jackie and Lisa can paint $\frac{11}{30}$ of the room per hour. To find how long it will take them to paint 1 room together, we take the reciprocal of their combined rate: $\text{Time to paint together} = \frac{1}{\frac{11}{30}} = \frac{30}{11}$

So, it will take them $\frac{30}{11}$ hours to paint the living room together.

Final Answer:

$\boxed{\frac{30}{11}} \text{ hours}$

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Would you like a more detailed explanation or have any questions? Here are some related questions:

1. How can we solve similar problems with more than two people working together?
2. What if one of the rates was negative, such as someone working against the task? How would that affect the result?
3. What happens if the work takes more than one day? How would we break up the total time into days and hours?
4. How can you apply this rate problem to work, travel, or other real-life scenarios?
5. Could you use this approach to figure out the rate of a machine working at a certain speed?

Tip: When solving rate problems, always remember to first find a common denominator before adding or subtracting fractions.