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Answer :
Let's solve the problem step-by-step to find out how many songs Miguel can buy with his gift card and which inequalities represent the situation.
1. Gift Card Details: Miguel has a gift card with a total amount of [tex]$25.
2. Costs Involved:
- Each song costs $[/tex]1.50.
- There is a one-time account activation fee of [tex]$1.00.
3. Setting Up the Inequality:
- Let \( m \) be the number of songs he can buy.
- The cost equation becomes: Total cost = Activation fee + Cost per song \(\times\) Number of songs.
- This can be written as:
\[
1.00 + 1.50m
\]
- Miguel wants to spend no more than the $[/tex]25 gift card amount, so set up the inequality:
[tex]\[
1.00 + 1.50m \leq 25
\][/tex]
4. Analyzing and Selecting Inequalities:
- We aim to find the inequalities that correctly represent this scenario from the given options.
Let's check the options:
- Option 1: [tex]\( 25 \geq 1 + 1.5m \)[/tex]
This option is valid as it directly matches our inequality: [tex]\( 1 + 1.5m \leq 25 \)[/tex].
- Option 2: [tex]\( 1 + 1.5m \leq 25 \)[/tex]
This is exactly the inequality we derived, so it's valid.
- Option 3: [tex]\( 1 + 1.5m < 25 \)[/tex]
This inequality suggests spending less than [tex]$25, which might not capture all possible exact scenarios, especially if the total cost equals exactly $[/tex]25.
- Option 4: [tex]\( 25 > 1 + 1.5m \)[/tex]
This implies that the total cost must be strictly less than [tex]$25, similar to Option 3, not capturing the equality case.
- Option 5: \( 1 + 1.5m \geq 25 \)
This one implies spending at least $[/tex]25, which contradicts Miguel's situation of not wanting to exceed the gift card amount.
Based on this analysis, the two correct inequalities are:
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
With these steps, Miguel can determine how many songs he can purchase without exceeding the gift card limit.
1. Gift Card Details: Miguel has a gift card with a total amount of [tex]$25.
2. Costs Involved:
- Each song costs $[/tex]1.50.
- There is a one-time account activation fee of [tex]$1.00.
3. Setting Up the Inequality:
- Let \( m \) be the number of songs he can buy.
- The cost equation becomes: Total cost = Activation fee + Cost per song \(\times\) Number of songs.
- This can be written as:
\[
1.00 + 1.50m
\]
- Miguel wants to spend no more than the $[/tex]25 gift card amount, so set up the inequality:
[tex]\[
1.00 + 1.50m \leq 25
\][/tex]
4. Analyzing and Selecting Inequalities:
- We aim to find the inequalities that correctly represent this scenario from the given options.
Let's check the options:
- Option 1: [tex]\( 25 \geq 1 + 1.5m \)[/tex]
This option is valid as it directly matches our inequality: [tex]\( 1 + 1.5m \leq 25 \)[/tex].
- Option 2: [tex]\( 1 + 1.5m \leq 25 \)[/tex]
This is exactly the inequality we derived, so it's valid.
- Option 3: [tex]\( 1 + 1.5m < 25 \)[/tex]
This inequality suggests spending less than [tex]$25, which might not capture all possible exact scenarios, especially if the total cost equals exactly $[/tex]25.
- Option 4: [tex]\( 25 > 1 + 1.5m \)[/tex]
This implies that the total cost must be strictly less than [tex]$25, similar to Option 3, not capturing the equality case.
- Option 5: \( 1 + 1.5m \geq 25 \)
This one implies spending at least $[/tex]25, which contradicts Miguel's situation of not wanting to exceed the gift card amount.
Based on this analysis, the two correct inequalities are:
- [tex]\( 25 \geq 1 + 1.5m \)[/tex]
- [tex]\( 1 + 1.5m \leq 25 \)[/tex]
With these steps, Miguel can determine how many songs he can purchase without exceeding the gift card limit.
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