Question: top-center: Aria is deciding between two parking garages. Garage A charges an initial fee of $7 to park plus $3 per hour. Garage B charges an initial fee of $4 to park plus $4 per hour. Let A represent the amount Garage A would charge if Aria parks for t hours, and let B represent the amount Garage B would charge if Aria parks for t hours. Write an equation for each situation, in terms of t, and determine the interval of hours parked, t, for which Garage A is cheaper than Garage B.
1. **State the problem:**
Aria wants to decide between two parking garages. Garage A charges $7 plus $3 per hour, and Garage B charges $4 plus $4 per hour. We want to write equations for the cost from each garage and find when Garage A is cheaper than Garage B.
2. **Write the cost equations:**
- Garage A cost: $$A = 7 + 3t$$
- Garage B cost: $$B = 4 + 4t$$
where $t$ is the number of hours parked.
3. **Set up inequality for Garage A being cheaper:**
$$7 + 3t < 4 + 4t$$
4. **Solve the inequality:**
Subtract $3t$ from both sides:
$$7 + \cancel{3t} < 4 + 4t - \cancel{3t}$$
which simplifies to
$$7 < 4 + t$$
Subtract 4 from both sides:
$$7 - 4 < 4 + t - 4$$
$$3 < t$$
5. **Interpret the solution:**
Garage A is cheaper than Garage B when the number of hours parked $t$ is greater than 3.
6. **Domain consideration:**
Since $t$ represents hours parked, $t \geq 0$. So the interval where Garage A is cheaper is:
$$t > 3$$ with $$t \geq 0$$, so effectively $$t \in (3, \infty)$$.
**Final answer:**
- $$A = 7 + 3t$$
- $$B = 4 + 4t$$
- Garage A is cheaper than Garage B when $$t > 3$$ hours.