Answer :
To solve this problem, we need to determine which inequalities represent the situation where Miguel uses a gift card to buy music. The gift card is worth [tex]$25, each song costs $[/tex]1.50, and there's a [tex]$1.00 account activation fee.
We will define \( m \) as the number of songs Miguel can purchase. The total cost for buying \( m \) songs is calculated by combining the account activation fee and the cost for the songs:
\[ \text{Total cost} = \$[/tex]1.00 + \[tex]$1.50 \times m \]
Miguel cannot spend more than what is on the gift card, which means the total cost has to be less than or equal to the value of the gift card, which is $[/tex]25. Thus, the inequality representing this constraint is:
[tex]\[ 1 + 1.5m \leq 25 \][/tex]
This tells us the maximum he can spend without exceeding the gift card amount.
Another way to express the same limiting factor is to say the gift card amount must be greater than or equal to the total cost:
[tex]\[ 25 \geq 1 + 1.5m \][/tex]
Additionally, Miguel can spend less than the entire value of the gift card. He doesn't have to use the entire amount, so another relevant inequality is:
[tex]\[ 1 + 1.5m < 25 \][/tex]
These inequalities cover both the scenarios of spending the full amount and spending less than the full amount on music. Therefore, the correct inequalities that represent this situation are:
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
- [tex]\( 1 + 1.5m < 25 \)[/tex]
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
These options allow us to model the problem accurately, ensuring Miguel stays within the budget provided by his gift card.
We will define \( m \) as the number of songs Miguel can purchase. The total cost for buying \( m \) songs is calculated by combining the account activation fee and the cost for the songs:
\[ \text{Total cost} = \$[/tex]1.00 + \[tex]$1.50 \times m \]
Miguel cannot spend more than what is on the gift card, which means the total cost has to be less than or equal to the value of the gift card, which is $[/tex]25. Thus, the inequality representing this constraint is:
[tex]\[ 1 + 1.5m \leq 25 \][/tex]
This tells us the maximum he can spend without exceeding the gift card amount.
Another way to express the same limiting factor is to say the gift card amount must be greater than or equal to the total cost:
[tex]\[ 25 \geq 1 + 1.5m \][/tex]
Additionally, Miguel can spend less than the entire value of the gift card. He doesn't have to use the entire amount, so another relevant inequality is:
[tex]\[ 1 + 1.5m < 25 \][/tex]
These inequalities cover both the scenarios of spending the full amount and spending less than the full amount on music. Therefore, the correct inequalities that represent this situation are:
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
- [tex]\( 1 + 1.5m < 25 \)[/tex]
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
These options allow us to model the problem accurately, ensuring Miguel stays within the budget provided by his gift card.
