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Answer :
To solve the problem of determining the inequalities that represent the situation of Miguel purchasing songs with his gift card, let's break it down step by step.
### Understanding the Problem:
- Miguel has a gift card with a total value of [tex]$25.
- Each song Miguel buys costs $[/tex]1.50.
- There's a one-time account activation fee of [tex]$1.00.
- We need to find inequalities that denote the maximum number of songs ($[/tex]m[tex]$) Miguel can purchase, taking into account these costs.
### Step-by-Step:
1. Set Up the Total Cost Equation:
- Total cost to buy $[/tex]m[tex]$ songs includes the activation fee and the cost for each song:
\[
\text{Total Cost} = 1 + 1.5m
\]
2. Express the Constraint:
- Miguel can spend up to $[/tex]25 (the amount on the gift card):
[tex]\[
1 + 1.5m \leq 25
\][/tex]
3. Write the Inequalities:
- Let's compare the total cost and the value of the gift card through inequalities:
- The total cost cannot exceed the gift card value:
[tex]\[
1 + 1.5m \leq 25 \quad \text{(this aligns with the inequality $25 \geq 1 + 1.5m$)}
\][/tex]
- Re-arranging gives us:
[tex]\[
25 > 1 + 1.5m \quad \text{(this inequality is also $25 > 1 + 1.5m$)}
\][/tex]
4. Evaluating Given Options:
- Option 1: [tex]\(25 \geq 1 + 15m\)[/tex] (incorrect as it's using [tex]$15m$[/tex] instead of [tex]$1.5m$[/tex])
- Option 2: [tex]\(1 + 15m < 25\)[/tex] (incorrect for the same reason as above)
- Option 3: [tex]\(25 > 1 + 1.5m\)[/tex] (correct as per our derived inequality)
- Option 4: [tex]\(1 + 1.5m \geq 25\)[/tex] (incorrect, it contradicts the situation)
- Option 5: [tex]\(1 + 15m \leq 25\)[/tex] (incorrect as it uses [tex]$15m$[/tex] instead of [tex]$1.5m$[/tex])
### Conclusion:
After evaluating the inequalities and the options given, the two valid inequalities that represent this situation are:
- Option 3: [tex]\(25 > 1 + 1.5m\)[/tex]
- Option 2: [tex]\(1 + 1.5m < 25\)[/tex]
These inequalities accurately represent the maximum number of songs Miguel can purchase with his gift card while accounting for both the song cost and the activation fee.
### Understanding the Problem:
- Miguel has a gift card with a total value of [tex]$25.
- Each song Miguel buys costs $[/tex]1.50.
- There's a one-time account activation fee of [tex]$1.00.
- We need to find inequalities that denote the maximum number of songs ($[/tex]m[tex]$) Miguel can purchase, taking into account these costs.
### Step-by-Step:
1. Set Up the Total Cost Equation:
- Total cost to buy $[/tex]m[tex]$ songs includes the activation fee and the cost for each song:
\[
\text{Total Cost} = 1 + 1.5m
\]
2. Express the Constraint:
- Miguel can spend up to $[/tex]25 (the amount on the gift card):
[tex]\[
1 + 1.5m \leq 25
\][/tex]
3. Write the Inequalities:
- Let's compare the total cost and the value of the gift card through inequalities:
- The total cost cannot exceed the gift card value:
[tex]\[
1 + 1.5m \leq 25 \quad \text{(this aligns with the inequality $25 \geq 1 + 1.5m$)}
\][/tex]
- Re-arranging gives us:
[tex]\[
25 > 1 + 1.5m \quad \text{(this inequality is also $25 > 1 + 1.5m$)}
\][/tex]
4. Evaluating Given Options:
- Option 1: [tex]\(25 \geq 1 + 15m\)[/tex] (incorrect as it's using [tex]$15m$[/tex] instead of [tex]$1.5m$[/tex])
- Option 2: [tex]\(1 + 15m < 25\)[/tex] (incorrect for the same reason as above)
- Option 3: [tex]\(25 > 1 + 1.5m\)[/tex] (correct as per our derived inequality)
- Option 4: [tex]\(1 + 1.5m \geq 25\)[/tex] (incorrect, it contradicts the situation)
- Option 5: [tex]\(1 + 15m \leq 25\)[/tex] (incorrect as it uses [tex]$15m$[/tex] instead of [tex]$1.5m$[/tex])
### Conclusion:
After evaluating the inequalities and the options given, the two valid inequalities that represent this situation are:
- Option 3: [tex]\(25 > 1 + 1.5m\)[/tex]
- Option 2: [tex]\(1 + 1.5m < 25\)[/tex]
These inequalities accurately represent the maximum number of songs Miguel can purchase with his gift card while accounting for both the song cost and the activation fee.
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