Answer :
Certainly! Let's tackle this problem step by step.
### Understanding the Problem
Miguel has a gift card worth [tex]$25. He wants to use this card to purchase songs, where each song costs $[/tex]1.50. Additionally, there is a [tex]$1.00 activation fee that he must pay regardless of how many songs he buys. We need to determine how many songs he can purchase without overspending the $[/tex]25 on his gift card.
### Setting Up the Inequality
We define [tex]\( m \)[/tex] as the number of songs Miguel can buy. Now, let's break down the costs:
1. Cost for songs: Each song costs [tex]$1.50, so the total cost for \( m \) songs is \( 1.5m \).
2. Activation fee: There is an additional $[/tex]1.00 fee that must be paid.
Therefore, the total cost of buying [tex]\( m \)[/tex] songs would be:
[tex]\[ \text{Total Cost} = 1 + 1.5m \][/tex]
Miguel can spend up to, but not more than, the [tex]$25 on his gift card. This sets up the inequality:
\[ 1 + 1.5m \leq 25 \]
### Confirming the Options
Now, we evaluate which inequalities correctly represent this situation from the given options.
1. Option 1: \( 1 + 1.5m \leq 25 \)
- This correctly represents the situation as explained above.
2. Option 2: \( 1 + 1.5m \geq 25 \)
- This does not make sense for the setup, as it implies spending more than or equal to $[/tex]25, but Miguel cannot exceed his gift card balance.
3. Option 3: [tex]\( 25 > 1 + 1.5m \)[/tex]
- This option can also be correct because it ensures that the total cost is less than [tex]$25, which means Miguel stays within his budget.
4. Option 4: \( 1 + 1.5m < 25 \)
- Similar to Option 3, it ensures the total cost is less than $[/tex]25, which makes this inequality valid for some scenarios.
5. Option 5: [tex]\( 25 \geq 1 + 1.5m \)[/tex]
- This is essentially the same as Option 1, just flipped around, and is also a valid representation of the budget constraint.
### Conclusion
The inequalities that can represent the situation of Miguel's song purchase are:
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
Both reflect his budget limit, ensuring he doesn't overspend his $25 gift card.
### Understanding the Problem
Miguel has a gift card worth [tex]$25. He wants to use this card to purchase songs, where each song costs $[/tex]1.50. Additionally, there is a [tex]$1.00 activation fee that he must pay regardless of how many songs he buys. We need to determine how many songs he can purchase without overspending the $[/tex]25 on his gift card.
### Setting Up the Inequality
We define [tex]\( m \)[/tex] as the number of songs Miguel can buy. Now, let's break down the costs:
1. Cost for songs: Each song costs [tex]$1.50, so the total cost for \( m \) songs is \( 1.5m \).
2. Activation fee: There is an additional $[/tex]1.00 fee that must be paid.
Therefore, the total cost of buying [tex]\( m \)[/tex] songs would be:
[tex]\[ \text{Total Cost} = 1 + 1.5m \][/tex]
Miguel can spend up to, but not more than, the [tex]$25 on his gift card. This sets up the inequality:
\[ 1 + 1.5m \leq 25 \]
### Confirming the Options
Now, we evaluate which inequalities correctly represent this situation from the given options.
1. Option 1: \( 1 + 1.5m \leq 25 \)
- This correctly represents the situation as explained above.
2. Option 2: \( 1 + 1.5m \geq 25 \)
- This does not make sense for the setup, as it implies spending more than or equal to $[/tex]25, but Miguel cannot exceed his gift card balance.
3. Option 3: [tex]\( 25 > 1 + 1.5m \)[/tex]
- This option can also be correct because it ensures that the total cost is less than [tex]$25, which means Miguel stays within his budget.
4. Option 4: \( 1 + 1.5m < 25 \)
- Similar to Option 3, it ensures the total cost is less than $[/tex]25, which makes this inequality valid for some scenarios.
5. Option 5: [tex]\( 25 \geq 1 + 1.5m \)[/tex]
- This is essentially the same as Option 1, just flipped around, and is also a valid representation of the budget constraint.
### Conclusion
The inequalities that can represent the situation of Miguel's song purchase are:
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
Both reflect his budget limit, ensuring he doesn't overspend his $25 gift card.
