Answer :
To solve the problem of finding which inequalities represent the situation where Miguel can buy songs with his [tex]$25 gift card, we need to consider a few things. Each song costs $[/tex]1.50, and there is a one-time account activation fee of [tex]$1.00. We want to determine how many songs, represented by \( m \), can be bought under these conditions.
### Step-by-Step Solution:
1. Understand the Equation:
- The total cost of buying songs includes a one-time activation fee plus the cost of each song.
- The cost equation is: \( \text{Total cost} = 1 + 1.5m \), where \( 1 \) is the activation fee, and \( 1.5m \) is the cost of \( m \) songs.
2. Set up the Inequality:
- Miguel can spend up to $[/tex]25. Therefore, the total cost should be less than or equal to [tex]$25. This is expressed as:
\[
1 + 1.5m \leq 25
\]
3. Rewriting the Inequality with Logical Equivalents:
- Rearranging and exploring equivalent and helpful forms gives us the inequalities:
- \( 1 + 1.5m \leq 25 \)
- \( 1 + 1.5m < 25 \)
- \( 25 \geq 1 + 1.5m \)
- \( 25 > 1 + 1.5m \)
### Verifying Given Options:
Considering the options given:
- \( 1 + 1.5m \leq 25 \): This is a correct representation, matches our initial understanding, and means the total spend should not exceed $[/tex]25.
- [tex]\( 1 + 1.5m \geq 25 \)[/tex]: This is not applicable since it suggests spending more than or equal to [tex]$25, which conflicts with our available budget.
- \( 25 > 1 + 1.5m \): This suggests the total cost is strictly less than $[/tex]25, making it a possible representation of spending within the budget.
- [tex]\( 1 + 1.5m < 25 \)[/tex]: This inequality allows for the cost to be less than [tex]$25, which is a valid representation for the budget constraint.
- \( 25 \geq 1 + 1.5m \): This also holds, akin to saying the budget can cover up to $[/tex]25 dollars.
### Conclusion:
The inequalities that correctly represent the situation are:
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
These both express the condition that Miguel's total spending, including the activation fee and song costs, should not exceed his $25 gift card balance.
### Step-by-Step Solution:
1. Understand the Equation:
- The total cost of buying songs includes a one-time activation fee plus the cost of each song.
- The cost equation is: \( \text{Total cost} = 1 + 1.5m \), where \( 1 \) is the activation fee, and \( 1.5m \) is the cost of \( m \) songs.
2. Set up the Inequality:
- Miguel can spend up to $[/tex]25. Therefore, the total cost should be less than or equal to [tex]$25. This is expressed as:
\[
1 + 1.5m \leq 25
\]
3. Rewriting the Inequality with Logical Equivalents:
- Rearranging and exploring equivalent and helpful forms gives us the inequalities:
- \( 1 + 1.5m \leq 25 \)
- \( 1 + 1.5m < 25 \)
- \( 25 \geq 1 + 1.5m \)
- \( 25 > 1 + 1.5m \)
### Verifying Given Options:
Considering the options given:
- \( 1 + 1.5m \leq 25 \): This is a correct representation, matches our initial understanding, and means the total spend should not exceed $[/tex]25.
- [tex]\( 1 + 1.5m \geq 25 \)[/tex]: This is not applicable since it suggests spending more than or equal to [tex]$25, which conflicts with our available budget.
- \( 25 > 1 + 1.5m \): This suggests the total cost is strictly less than $[/tex]25, making it a possible representation of spending within the budget.
- [tex]\( 1 + 1.5m < 25 \)[/tex]: This inequality allows for the cost to be less than [tex]$25, which is a valid representation for the budget constraint.
- \( 25 \geq 1 + 1.5m \): This also holds, akin to saying the budget can cover up to $[/tex]25 dollars.
### Conclusion:
The inequalities that correctly represent the situation are:
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
These both express the condition that Miguel's total spending, including the activation fee and song costs, should not exceed his $25 gift card balance.
