Unraveling First-Degree Inequalities 

Entering the Discovery Portal

Did you know that in 1832, the French mathematician Évariste Galois, at just 20 years old, wrote letters on the night before his fatal duel, developing fundamental theories for modern algebra, including concepts related to groups? This historical tragedy not only emphasizes youthful genius but also the depth of mathematics in solving complex problems. 類✨

Quizz:  Have you ever stopped to think about how your life would be if each of your decisions was determined by a linear inequality? Imagine solving everyday problems, like planning your time or your money, using only mathematics . What is the difference between an equation and an inequality, and how does this apply to your daily life?

Exploring the Surface

First-degree inequalities are like the slightly more complicated sister of equations. While equations seek to find a specific value, inequalities help us understand a range of possible values for a variable.  They are powerful tools that allow us to make predictions, carry out planning, and make decisions grounded in real data and varied contexts.

First-degree inequalities involve linear expressions, which are mathematical formulas representing linear relationships between variables. This type of inequality is widely used to solve practical problems, such as determining whether a budget is sufficient to cover certain expenses or whether the available time is adequate to complete a task.  Simply put, inequalities are super useful for situations where we’re not looking for just an exact result, but a range of possibilities.

In this chapter, we will explore deeply what first-degree inequalities are, how to solve them, and how to apply them in various contexts.  Unraveling these concepts not only enhances your mathematical skills but also improves your ability to solve everyday problems in a logical and structured way. Get ready to embark on this journey and discover how mathematics can be a great ally in your life! 

The Wonderful World of Inequalities 

Let's start with the basics: what exactly is a first-degree inequality? Imagine you are in line for a roller coaster and you can only get on if you are 1.50m or taller. Ah, those heights! An equation would be like knowing the exact number of your height; an inequality, on the other hand, is like knowing you are within a range of heights (1.50m or taller). In mathematical terms, an inequality is an expression that shows that two things are not equal and that one is greater or less than the other. Interestingly, understanding this can actually help avoid lines, but I digress.

What does 'first degree' mean? Think of a thermometer; 'first degree' in mathematics means we only have one variable raised to the first power (easy, no panic!). So, when we say 'first-degree inequality', we are dealing with inequalities involving variables that are not raised to squares, cubes, or any other far-off mathematical planet.

Here's a practical example: do you know those moments when you need to decide whether it’s worth buying two packs of snacks or if your money is not enough? If 2 snacks cost, for example, 5 reais each, you can use an inequality to determine if your 15 reais are sufficient (spoiler: they are). The inequality would be 2x ≤ 15, where 'x' is the price of a snack. Solve this and find out that you can buy both packs and still pay that mysterious refund of one real you lost during recess! 

Proposed Activity: Snack Challenge 

Take a photo of something everyday that could be solved with a first-degree inequality, such as a snack bar menu with prices. Share it in the class WhatsApp group and explain how you would solve the problem with an inequality. We will raffle a better explanation to highlight in class! 

The Great Adventure of Solving Inequalities! ️‍♂️

Get ready, aim, solve! Solving an inequality is a bit like being an old-time detective, only with fewer dramatic shadows and more numbers. Suppose you want to dissolve a mystery: 2x - 4 > 6. First, you need to eliminate the suspect (that is, the number) that’s obstructing the investigation on the side with the unknown 'x'. In our case, we add 4 to both sides: 2x > 10.

The next step is to divide everything by the coefficient of the variable, but be careful! If you multiply or divide by a negative number, things get messier than a movie chase scene: you need to change the sign of the inequality. In our example: 2x > 10. Dividing by 2, we have x > 5. Ta-da, you found the answer! But there’s no suspense music, just the joy of knowing that x needs to be greater than 5.

Let's complicate it a bit just for fun? Imagine that 3(x - 2) ≤ 9. First, distribute the 3: 3x - 6 ≤ 9. Now add 6 to both sides because math loves this zen balancing: 3x ≤ 15. To finish, divide by 3 and voilà: x ≤ 5. Super simple, no suspense theme and, certainly, no sinister mustache villain! 

Proposed Activity: Inequality Detective Challenge 

Create an inequality based on a real problem you face in your daily life, like managing your time between school and your favorite series! Post your inequality in the class forum and solve it. Let’s see who has the most unique challenges! 

Life is Full of Inequalities 

You know that moment in a game where you have life points and need to decide whether to spend them all on a potion or save some for armor? Exactly, you might be using inequalities without realizing it! If you have 20 life points and each potion heals 8 points, the inequality 8x ≤ 20 helps you figure out how many potions you can buy without bursting your stock. In this case, x ≤ 2.5 (better buy two potions). Who would have thought math could save you in games and in real life?

Let's give a curious example: you want to organize a party, but you have a limited budget (of course, the sad reality). If each balloon costs 3 reais and each treat 2 reais, and you have 50 reais in your pocket, the inequality 3x + 2y ≤ 50 will guide you. Similar to a tour guide, only less sophisticated. It means that if you want 10 balloons (who doesn't want balloons?), that costs 30 reais, leaving 20 reais for up to 10 treats. Delicious math!

So, next time you're caught in a battle between the amount of rice for a family lunch versus the time in the microwave, remember: inequalities are everywhere. And when you need to calculate how many minutes of study you need to ensure gaming time without guilt? Yes, once again, inequalities to the rescue! +=❤

Proposed Activity: Daily Inequality Dilemma 樂

Write down an example of how you would use an inequality in a daily dilemma, like spending your allowance or dividing your time. Post it in the class forum and compare your situations with your classmates. It’s math in real life! 

Ninja Inequality: Tricks and Tips 屢

It’s time to become a ninja master of inequalities! And like any good ninja training, practice makes perfect. First trick: never, ever, mess with the inequality sign without a good reason, especially when multiplying or dividing by negative numbers – it’s like trying to hit a fly with a nunchaku; it might go wrong!

Second trick: always simplify whenever you can. If you find yourself with something like 5x + 7 < 2x + 13, first get rid of the excess (kind of like cleaning your room before your mom sees). Subtract 2x from both sides: 3x + 7 < 13. Then, subtract 7 from both sides: 3x < 6. Finally, divide by 3 and there you go – x < 2. It seems like magic, but it’s just math!

The third and final trick is to watch out for traps. For example, beware of parentheses! They are not just decoration. If you see something like 2(3x - 4) > 8, distribute the 2 first: 6x - 8 > 8. Then, add 8 to both sides: 6x > 16. Finally, divide by 6: x > 16/6 or x > 8/3. With these tricks, you'll be prepared for any unequal situation in life or if you need to face a cyber dojo master! 屢

Proposed Activity: Ninja Inequality Challenge 朗

Practice these tricks by solving the following inequality: -2x + 5 > 1. Post the solution in the WhatsApp group and see who solves it the fastest and correctly! May the best ninja win! 朗

Creative Studio

Summary of the most important points of the Chapter in poem format.

Reflections

• List of reflections on the Chapter.

Your Turn...

Reflection Journal

Write and share with your class three of your own reflections on the topic.

Systematize

Create a mind map on the topic studied and share it with your class.

Conclusion

Congratulations on reaching the end of this chapter and becoming a true detective of inequalities! Now you know what first-degree inequalities are, how to solve them, and most importantly, how to apply them in your daily life.  This knowledge is not only theoretical; it is a powerful tool you can use to make more informed and logical decisions in various situations.

As you prepare for our Active Class, revisit the examples and activities we explored here. Consider sharing your discoveries and thoughts on social media, and who knows, you might become a math influencer! In our class, we will delve even deeper into these concepts with practical and exciting activities that promise to transform your way of viewing mathematics. Get ready, the challenge has just begun, and you are already one step ahead! 