College

Miguel can use all or part of his [tex]$\$25[/tex] gift card to make a music purchase. Each song costs [tex]$1.50[/tex], and there is a [tex]$\$1.00[/tex] per account activation fee.

Which inequalities can represent this situation if [tex]m[/tex] is the number of songs he can buy? Select two options.

A. [tex]1 + 1.5m \leq 25[/tex]
B. [tex]1 + 1.5m \geq 25[/tex]
C. [tex]25 > 1 + 1.5m[/tex]
D. [tex]1 + 1.5m < 25[/tex]
E. [tex]25 \geq 1 + 1.5m[/tex]

Answer :

Sure! Let's work through the problem step-by-step to find the correct inequalities representing the situation.

Miguel has a [tex]$25 gift card, and he wants to use it to buy songs, each costing $[/tex]1.50. There's also a [tex]$1.00 account activation fee. Let $[/tex]m[tex]$ be the number of songs Miguel can buy.

Step 1: Determine the Total Cost Equation

The total cost for Miguel buying $[/tex]m[tex]$ songs will include both the cost of the songs and the activation fee.

- Activation fee: $[/tex]1.00
- Cost of [tex]$m$[/tex] songs: [tex]$1.50 \times m$[/tex]

Therefore, the total cost [tex]$C$[/tex] can be written as:
[tex]\[ C = 1.00 + 1.50 \times m \][/tex]

Step 2: Set Up the Inequality

Since Miguel cannot spend more than the [tex]$25 on his gift card, the total cost must be less than or equal to $[/tex]25. Therefore, we set up the following inequality:

[tex]\[ 1 + 1.5m \leq 25 \][/tex]

This inequality means that the sum of the account activation fee and the total song cost should not exceed the [tex]$25 gift card balance.

Step 3: Understand the Equivalent Form

The above inequality can also be written with the sides flipped:

\[ 25 \geq 1 + 1.5m \]

This means the same thing: the total money on the gift card ($[/tex]25) is greater than or equal to the total cost of the songs and activation fee.

Conclusion: Identify the Correct Options

Based on the steps above, the two inequalities that describe the situation correctly are:

1. [tex]\( 1 + 1.5m \leq 25 \)[/tex]
2. [tex]\( 25 \geq 1 + 1.5m \)[/tex]

These inequalities ensure that the cost of buying [tex]$m$[/tex] songs plus the activation fee does not exceed [tex]$25.

Therefore, the correct selections are:
- $[/tex]1+1.5 m \leq 25[tex]$
- $[/tex]25 \geq 1+1.5 m$