Parkcity Media Article

Gift Card Music Purchase: Which Inequality Is Right?

Dec 17, 2025 by Andrew McMorgan 53 views

Hey guys, let's dive into a super common situation that many of us run into: using a gift card for some sweet music! Miguel's got a $25 gift card, and he's itching to load up on some new tunes. But here's the catch – each song has a price tag of $1.50, and there's also a one-time $1.00 fee just to activate his account. So, the big question is, ***which inequality can actually represent the number of songs, represented by '$ m

, that Miguel can buy with his gift card?*** This is all about translating a real-world scenario into a mathematical expression, and trust me, it's a skill that comes in handy way more often than you might think. We're going to break down the problem, look at the options, and figure out the one that truly captures Miguel's music-buying power.

Understanding the Variables and Costs

Alright, let's get down to brass tacks, shall we? We've got a few key players in this scenario. First off, we have Miguel's total budget, which is a sweet $25 gift card. This is the absolute maximum he can spend. Next up, we have the cost per song. Each track Miguel wants to snag will set him back $1.50. Now, this cost multiplies with every song he buys. If he buys 1 song, it's $1.50. If he buys 2, it's $1.50 x 2, and so on. This is where our variable, '$ m $', comes into play. '$ m

represents the number of songs Miguel decides to purchase. So, the total cost of the songs themselves will be

1.50 multiplied by '

m

, or $1.50m$ . But wait, there's a little extra fee! Before Miguel can even start downloading his favorite jams, there's a $1.00 per account activation fee. This is a fixed cost, meaning it doesn't change no matter how many songs he buys. It's a one-and-done charge. So, the total amount Miguel will spend isn't just the cost of the songs; it's the cost of the songs plus that activation fee. This is a crucial point, guys, because we need to account for all the money going out of that gift card. The total cost will therefore be $1.50m + $1.00$ . Now, remember that $25 gift card? Miguel can use all or part of it. This means his total spending must be less than or equal to the $25 he has available. So, the total cost, $1.50m +

1.00

, must be less than or equal to $25. This forms the basis of our inequality. It's like setting a boundary on how much he can splurge on his music collection. We're essentially saying, 'The money spent on songs plus the activation fee cannot exceed the gift card's value.' Keep these numbers and their roles in mind as we move on to evaluating the inequality options.

Analyzing the Inequality Options

Now that we've got a solid grasp on the costs involved – the variable cost per song and the fixed activation fee – let's look at the potential inequalities that could represent Miguel's situation. Remember, ' $m$ ' stands for the number of songs, and he has a total of $25 to spend. The total cost is the sum of the cost of the songs ( $1.50m$ ) and the activation fee ( $1.00$ ). This total cost must be less than or equal to his gift card limit of $25. So, we're looking for an inequality that looks something like: $1.50m + 1.00 ext{ (or } 1 ext{) } ext{ extless or equal to } 25$ . Let's consider the options presented, even if they aren't fully listed in the prompt, and think about why some might be right and others definitively wrong.

Option A, for instance, might look something like $1 + 15m ext{ extless or equal to } 25$ or $1.50m + 1 ext{ extless or equal to } 25$ . We need to carefully check if the coefficients and constants match our understanding. The $1.00 activation fee should be a standalone term, not multiplied by '$ m

, because it's a one-time charge. The

1.50 per song cost *must* be multiplied by '

m

. So, $1.50m + 1 ext{ extless or equal to } 25$ seems to be on the right track. Let's think about common mistakes people make. Sometimes, the activation fee is mistakenly multiplied by '

m

'. This would lead to an inequality like $1.00m + 1.50 ext{ extless or equal to } 25$ , which is incorrect because it implies the activation fee changes with the number of songs. Another common error is mixing up the coefficients. If someone wrote $15m + 1 ext{ extless or equal to } 25$ , they might have accidentally dropped the decimal point for the song cost, making each song cost $15.00, which is way too much for a song and clearly wrong. We also need to ensure the inequality symbol is correct. Since Miguel can use all or part of his gift card, his spending can be equal to $25 or less than

25. Thus, the 'less than or equal to' symbol (

ext{ extless or equal to}

) is the correct one to use, not just 'less than' (

ext{ extless}$). If the inequality were strictly 'less than', it would imply he couldn't spend the exact amount of $25, which isn't the case here. So, we're hunting for the inequality that accurately reflects $1.50m$ for the songs, $+ 1$ for the activation fee, and $ ext{ extless or equal to } 25$ for the total budget.

Solving for the Number of Songs

Okay, so we've identified the correct inequality that represents Miguel's music purchasing power. It's the one that accurately combines the cost of the songs and the activation fee, keeping it all within the bounds of his $25 gift card. The inequality that best describes this situation is $1.50m + 1 ext{ extless or equal to } 25$ . Now, while the question only asks which inequality can represent this situation, it's super useful and satisfying to actually solve it to see just how many songs Miguel can buy. This helps confirm our inequality is logical and gives us a concrete answer to Miguel's musical quest.

Let's tackle this step-by-step. Our goal is to isolate ' $m$ ', the number of songs. First, we need to get rid of that pesky $1.00 activation fee on the left side of the inequality. To do this, we subtract $1.00$ from both sides. Remember, whatever you do to one side of an inequality, you must do to the other to keep it balanced.

So, we have:

$1.50m + 1 ext{ extless or equal to } 25$

Subtract 1 from both sides:

$1.50m + 1 - 1 ext{ extless or equal to } 25 - 1$

This simplifies to:

$1.50m ext{ extless or equal to } 24$

Awesome! Now we've isolated the cost of the songs. The next step is to figure out how many songs ' $m$ ' can be. Since $1.50 is multiplied by '$ m

, we need to divide both sides by

1.50

to get '

m

' by itself.

$1.50m / 1.50 ext{ extless or equal to } 24 / 1.50$

Now, let's do the division. $24$ divided by $1.50$ is:

$24 / 1.50 = 16$

So, the inequality becomes:

$m ext{ extless or equal to } 16$

What does this tell us, guys? It means that Miguel can buy 16 songs or fewer. This is the maximum number of songs he can possibly purchase while staying within his $25 gift card limit, including that activation fee. It’s great to see how the math works out and gives us a clear, actionable number. This solution confirms that our chosen inequality was indeed the correct representation of the problem. If we were to buy 16 songs, the total cost would be ($1.50 imes 16) + 1 = 24 + 1 = $25. Exactly on budget! If he tried to buy 17 songs, the cost would be ($1.50 imes 17) + 1 = 25.50 + 1 = $26.50, which is over his gift card limit. So, yes, $m ext{ extless or equal to } 16$ is the concrete outcome, derived from the correct inequality representing the situation.

Conclusion: Choosing the Right Inequality

So, after breaking down all the costs and understanding how inequalities work, we've arrived at the definitive answer for which inequality best represents Miguel's music purchase scenario. Remember, we have a total budget of $25, a per-song cost of $1.50 (multiplied by the number of songs, '$ m

), and a one-time activation fee of $1.00. The total spending must be less than or equal to the gift card amount. Therefore, the correct inequality is $1.50m + 1 ext{ extless or equal to } 25$ . This inequality accurately captures all aspects of the problem: the variable cost of the songs, the fixed activation fee, and the budget constraint. It's crucial to get these components right – ensuring the

1.50

is multiplied by '

m

' and that the

1.00

is a separate, added cost, and using the 'less than or equal to' sign because Miguel can spend up to the full amount. Any deviation from this structure, like multiplying the activation fee by '

m

' or using the wrong inequality symbol, would misrepresent the situation and lead to incorrect conclusions about how much music Miguel can actually buy. It’s this careful attention to detail in translating words into mathematical symbols that makes problem-solving so rewarding. Keep practicing, and you'll become inequality wizards in no time, guys!