Answer :
Sure, let's break down the solution step-by-step.
### Problem Breakdown:
- Miguel has a [tex]$25 gift card.
- Each song costs $[/tex]1.50.
- There is also a [tex]$1.00 account activation fee.
We need to establish how many songs, \( m \), Miguel can purchase under these conditions.
### Steps to Find the Inequalities:
1. Total Cost Calculation:
The total cost for Miguel to activate his account and purchase \( m \) songs can be represented as:
\[
\text{Total Cost} = 1.00 + 1.50m
\]
2. Setting Up Inequalities:
Since Miguel’s total expenditure must not exceed the amount on the gift card ($[/tex]25), we set up the inequality:
[tex]\[
1.00 + 1.50m \leq 25
\][/tex]
This means the total cost should be less than or equal to [tex]$25.
3. Alternative Representation:
Similarly, we can express the condition that the total cost should not exceed $[/tex]25 as:
[tex]\[
25 \geq 1.00 + 1.50m
\][/tex]
This is simply the same inequality written in reverse order.
4. Exploring Other Options:
We can also consider the inequalities where the total cost is strictly less than [tex]$25, i.e.:
\[
1.00 + 1.50m < 25
\]
Or equivalently:
\[
25 > 1.00 + 1.50m
\]
### Conclusion:
Based on these considerations, we derived four possible inequalities that can represent the situation:
- \(1 + 1.5m \leq 25\)
- \(25 \geq 1 + 1.5m\)
- \(1 + 1.5m < 25\)
- \(25 > 1 + 1.5m\)
From these, the two options that correctly represent the situation are:
1. \(1 + 1.5m \leq 25\)
2. \(25 \geq 1 + 1.5m\)
These two inequalities are essentially saying the same thing, hence they both are correct.
### Incorrect Options:
- \(1 + 1.5m \geq 25\) is incorrect as it implies the total cost exceeds the gift card amount.
- \(25 \leq 1 + 1.5m\) is also incorrect for the same reason.
- \(1 + 1.5m < 25\) is correct, but it doesn’t represent the most inclusive condition (since he could spend exactly $[/tex]25).
So, the inequalities that can represent this situation correctly are:
- [tex]\(1 + 1.5m \leq 25\)[/tex]
- [tex]\(25 \geq 1 + 1.5m\)[/tex]
- [tex]\(1 + 1.5m < 25\)[/tex]
- [tex]\(25 > 1 + 1.5m\)[/tex]
Among these, the two options that we initially focused on were [tex]\(1 + 1.5m \leq 25\)[/tex] and [tex]\(25 \geq 1 + 1.5m\)[/tex].
### Problem Breakdown:
- Miguel has a [tex]$25 gift card.
- Each song costs $[/tex]1.50.
- There is also a [tex]$1.00 account activation fee.
We need to establish how many songs, \( m \), Miguel can purchase under these conditions.
### Steps to Find the Inequalities:
1. Total Cost Calculation:
The total cost for Miguel to activate his account and purchase \( m \) songs can be represented as:
\[
\text{Total Cost} = 1.00 + 1.50m
\]
2. Setting Up Inequalities:
Since Miguel’s total expenditure must not exceed the amount on the gift card ($[/tex]25), we set up the inequality:
[tex]\[
1.00 + 1.50m \leq 25
\][/tex]
This means the total cost should be less than or equal to [tex]$25.
3. Alternative Representation:
Similarly, we can express the condition that the total cost should not exceed $[/tex]25 as:
[tex]\[
25 \geq 1.00 + 1.50m
\][/tex]
This is simply the same inequality written in reverse order.
4. Exploring Other Options:
We can also consider the inequalities where the total cost is strictly less than [tex]$25, i.e.:
\[
1.00 + 1.50m < 25
\]
Or equivalently:
\[
25 > 1.00 + 1.50m
\]
### Conclusion:
Based on these considerations, we derived four possible inequalities that can represent the situation:
- \(1 + 1.5m \leq 25\)
- \(25 \geq 1 + 1.5m\)
- \(1 + 1.5m < 25\)
- \(25 > 1 + 1.5m\)
From these, the two options that correctly represent the situation are:
1. \(1 + 1.5m \leq 25\)
2. \(25 \geq 1 + 1.5m\)
These two inequalities are essentially saying the same thing, hence they both are correct.
### Incorrect Options:
- \(1 + 1.5m \geq 25\) is incorrect as it implies the total cost exceeds the gift card amount.
- \(25 \leq 1 + 1.5m\) is also incorrect for the same reason.
- \(1 + 1.5m < 25\) is correct, but it doesn’t represent the most inclusive condition (since he could spend exactly $[/tex]25).
So, the inequalities that can represent this situation correctly are:
- [tex]\(1 + 1.5m \leq 25\)[/tex]
- [tex]\(25 \geq 1 + 1.5m\)[/tex]
- [tex]\(1 + 1.5m < 25\)[/tex]
- [tex]\(25 > 1 + 1.5m\)[/tex]
Among these, the two options that we initially focused on were [tex]\(1 + 1.5m \leq 25\)[/tex] and [tex]\(25 \geq 1 + 1.5m\)[/tex].
