Dexter's Photography Earnings Representing Earnings With Inequalities

Hey guys! Let's dive into a super practical math problem today, one that photographers (or anyone earning within a range) can totally relate to. We're going to break down how to represent Dexter's photography session earnings using inequalities. It might sound a bit intimidating, but trust me, it's simpler than you think! This article will walk you through the problem step-by-step, ensuring you grasp the concept of inequalities and how they apply to real-world scenarios. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the inequalities themselves, let's make sure we fully understand the situation. Dexter, our awesome photographer, has a specific range for his earnings per session. He makes no less than $50, meaning he earns at least $50, and no more than $100, meaning his earnings cap out at $100. Our mission is to find the inequality that perfectly captures this earning range. To really nail this, let's break down those key phrases: "no less than" and "no more than." "No less than" indicates that the earnings can be equal to or greater than the specified amount. Think of it as a minimum threshold. On the flip side, "no more than" means the earnings can be equal to or less than the specified amount, setting a maximum limit. Visualizing this range is super helpful. Imagine a number line. Dexter's earnings start at $50 (or at least that much) and go all the way up to $100 (but not exceeding it). This gives us a clear picture of a contained range, where his earnings fall somewhere within these two dollar amounts. Now that we've painted a clear picture of Dexter's earning situation, we are well-prepared to translate this into the language of inequalities. We'll explore the symbols and their meanings, and then match them up with the correct representation of Dexter's earnings. Stick with me, and you'll be an inequality whiz in no time!

Decoding Inequalities: The Symbols

Okay, so we know Dexter's earnings fall between $50 and $100, but how do we write that down in math terms? That's where inequality symbols come in! These symbols are like a secret code that tells us the relationship between different values. Let's crack the code, shall we? The main players in the inequality symbol world are:

• > (greater than): This symbol means one value is larger than another. For example, 5 > 3 means 5 is greater than 3.
• < (less than): This symbol is the opposite, indicating one value is smaller than another. For instance, 2 < 7 means 2 is less than 7.
• ≥ (greater than or equal to): This is where things get a little more nuanced. This symbol means a value is either larger than or equal to another. So, x ≥ 4 means x can be 4 or any number bigger than 4.
• ≤ (less than or equal to): You guessed it! This symbol means a value is either smaller than or equal to another. For example, y ≤ 10 means y can be 10 or any number smaller than 10. Understanding these symbols is absolutely crucial for translating word problems into mathematical inequalities. Each symbol represents a different type of relationship, and choosing the right one is key to accurately representing the situation. Let's go back to Dexter's earnings. He earns "no less than" $50. Which symbol do you think best captures that idea? We need a symbol that includes the possibility of earning exactly $50, as well as earning more. The "greater than or equal to" symbol (≥) fits the bill perfectly! Similarly, for the "no more than" $100 part, we need a symbol that allows for earning exactly $100 or less. The "less than or equal to" symbol (≤) is our winner here. Now that we've deciphered the inequality symbols, we're ready to put them together and build the inequality that represents Dexter's photography session earnings.

Building the Inequality for Dexter's Earnings

Alright, we've got the problem, we've got the symbols, now let's put it all together and build the inequality! Remember, we want to represent Dexter's earnings, which we'll call "e", as being between $50 and $100. He earns no less than $50, which means his earnings are greater than or equal to $50. We can write that as: e ≥ 50. This part of the inequality sets the lower bound for his earnings. He also earns no more than $100, meaning his earnings are less than or equal to $100. We can write that as: e ≤ 100. This sets the upper bound for his earnings. Now, how do we combine these two inequalities into one? This is where the concept of a compound inequality comes in. A compound inequality is simply two inequalities joined together, usually using the word "and." In this case, we need to say that "e is greater than or equal to 50 and e is less than or equal to 100." There's a neat way to write this in a more compact form. We can combine the two inequalities like this: 50 ≤ e ≤ 100. This single inequality says it all! It means that "e" (Dexter's earnings) is greater than or equal to 50 and less than or equal to 100. It beautifully captures the range of his possible earnings in one concise statement. Think of it as a mathematical sandwich, with 50 and 100 as the bread and "e" as the delicious filling in between. This is the power of inequalities – they allow us to express ranges and relationships in a clear and efficient way. Now, let's take a look at the answer choices and see which one matches our perfectly crafted inequality.

Analyzing the Answer Choices

We've done the hard work of understanding the problem and building the correct inequality. Now comes the satisfying part – finding the matching answer! Let's revisit the inequality we came up with: 50 ≤ e ≤ 100. This means Dexter's earnings ("e") are between $50 and $100, inclusive. Now, let's break down the answer choices and see which one lines up:

A. e ≥ 50 or e ≤ 100: This option uses the word "or," which means either inequality can be true. While e ≥ 50 is correct (Dexter earns at least $50), e ≤ 100 is also correct (Dexter earns no more than $100). However, the "or" makes this too broad. It would include values outside of Dexter's earning range. For instance, if e = $40, then it's not true, so "or" is incorrect.

B. e > 50 or e < 100: This option is similar to A, but it uses "greater than" and "less than" instead of "greater than or equal to" and "less than or equal to." This means it doesn't include the endpoints of the range ($50 and $100). Plus, the "or" makes it too broad, just like in option A.

C. *50 100: This is the winner! It perfectly captures the idea that Dexter's earnings are greater than or equal to $50 and less than or equal to $100. It's the mathematical sandwich we talked about earlier, keeping Dexter's earnings neatly within the correct range.

So, there you have it! By carefully analyzing the problem, understanding inequality symbols, and building the correct inequality, we were able to confidently identify the correct answer. This skill of translating real-world scenarios into mathematical expressions is super valuable, not just in math class, but in everyday life too.

Real-World Applications of Inequalities

You might be thinking, "Okay, I solved Dexter's photography problem, but when will I ever use inequalities in the real world?" Well, guys, the truth is, inequalities are everywhere! They pop up in all sorts of situations, from managing your budget to understanding speed limits. Let's explore some real-world scenarios where inequalities come to the rescue. Imagine you're planning a party and have a budget of $200. You need to figure out how many pizzas you can order. If each pizza costs $15, you can use an inequality to represent the situation: 15x ≤ 200 (where x is the number of pizzas). Solving this inequality tells you the maximum number of pizzas you can buy without busting your budget. Inequalities are also super useful in understanding things like speed limits. A speed limit sign that says "65 mph" is actually an inequality in disguise! It means your speed (let's call it "s") must be less than or equal to 65 mph: s ≤ 65. Breaking this inequality could lead to a speeding ticket, so it's definitely a real-world application! In the world of finance, inequalities help us understand investment returns, loan interest rates, and all sorts of financial calculations. For example, if you want to earn at least $1000 in interest from an investment, you can use an inequality to figure out how much money you need to invest. Even in everyday situations like grocery shopping, inequalities can help you compare prices and make sure you're getting the best deal. If you want to spend no more than $5 on a box of cereal, you're essentially setting up an inequality in your head. The power of inequalities lies in their ability to represent constraints, limits, and ranges. They help us make informed decisions, plan effectively, and understand the world around us. So, the next time you see a "minimum purchase" sign or a "maximum occupancy" notice, remember that inequalities are at play!

Conclusion: Mastering Inequalities

We've reached the end of our inequality adventure, and what a journey it's been! We started with Dexter's photography earnings, broke down the language of inequalities, built the perfect inequality to represent his earning range, and even explored real-world applications. Hopefully, you've gained a solid understanding of what inequalities are and how they work. The key takeaway here is that inequalities are not just abstract mathematical concepts; they are powerful tools for representing real-world situations involving ranges, limits, and constraints. By mastering inequalities, you're not just acing math problems; you're developing critical thinking skills that will serve you well in various aspects of life. Remember the key steps we took: First, we carefully analyzed the problem to understand the relationships between the values. Then, we translated the words into mathematical symbols, using the correct inequality signs. Next, we combined the inequalities to create a comprehensive representation of the situation. Finally, we applied our understanding to solve the problem and make informed decisions. So, go forth and conquer the world of inequalities! Practice applying them to different scenarios, and you'll be amazed at how versatile and useful they can be. And remember, just like Dexter with his photography, you too can capture the world of math with the right tools and techniques! Keep practicing, keep exploring, and keep those mathematical gears turning!