Solving Inequalities Step By Step Identifying Properties Used In 7x + 4 < 46

Hey guys! Today, we're diving into the world of inequalities and how to solve them step-by-step. We'll be focusing on identifying the specific properties used at each stage of the solution process. To make things crystal clear, we'll take a look at the inequality $7x + 4 < 46$ and break down each step, explaining the mathematical magic behind it. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let's take a moment to understand what we're dealing with. An inequality, unlike an equation, doesn't state that two expressions are exactly equal. Instead, it shows a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. In our case, we have $7x + 4 < 46$, which means the expression $7x + 4$ is less than 46. Our goal is to isolate the variable x on one side of the inequality to determine the range of values that satisfy this condition. We will achieve this by carefully applying properties of inequality, ensuring that each step maintains the truth of the original statement. It is very important to understand that when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign flips, this crucial detail is rooted in the nature of negative numbers and their effect on ordering.

Step 1 Given

The Foundation of Our Journey

The first step in any mathematical problem, especially when solving inequalities, is simply stating the given information. In our case, the given inequality is $7x + 4 < 46$. This step might seem trivial, but it's crucial because it sets the stage for everything that follows. It's like laying the foundation of a house – you can't build anything solid without it. By explicitly stating what we're starting with, we ensure clarity and avoid any potential confusion down the line. Think of it as writing down the rules of the game before you start playing. We have to know where we’re starting from, so we can track every move we make. This initial step isn't about applying any mathematical operation; it's about acknowledging the problem we're setting out to solve. It provides a reference point, allowing us to trace our steps back if needed and verify the correctness of our solution. So, while it may seem straightforward, this initial declaration is an indispensable part of the problem-solving process. Remember, in mathematics, precision and clarity are key, and starting with the given information is the first step towards achieving both. It's like the first line of a well-written essay or the opening scene of a captivating movie – it sets the tone and direction for what's to come. Understanding this foundational aspect will help you approach more complex problems with confidence and ensure that you never lose sight of your starting point. This simple act of stating the given is a cornerstone of mathematical rigor, ensuring that every step we take is firmly grounded in the problem itself.

Step 2 $7x + 4 < 46$

Restating the Inequality The Starting Line

In the second step, we simply rewrite the inequality $7x + 4 < 46$. You might be wondering, “Why bother? We already have it!” Well, this step serves as a bridge between the "Given" statement and the actual manipulation of the inequality. It's like re-reading the question in a test to make sure you fully understand what's being asked before you start solving. It provides a clear visual transition, ensuring that we're not skipping any mental steps. While it might appear redundant, this reiteration is a good practice, especially when dealing with more complex problems. It helps prevent errors by reinforcing the starting point in our minds. Think of it as a quick pit stop in a race, just to check all systems are go before accelerating forward. We're essentially confirming that we're still working with the same inequality and haven't made any accidental changes or misinterpretations. This step also has a psychological benefit; it provides a moment to pause and mentally prepare for the upcoming steps. It's a chance to double-check our understanding and make sure we're ready to apply the necessary operations. In essence, restating the inequality is a simple yet effective way to maintain clarity and accuracy in our problem-solving process. It's a small step that can make a big difference in ensuring we arrive at the correct solution. So, don't underestimate the power of this seemingly trivial step – it's an important part of the mathematical journey.

Step 3 $7x < 42$

Applying the Subtraction Property of Inequality The Key Move

Now we're getting into the real action! In step 3, we transform the inequality $7x + 4 < 46$ into $7x < 42$. The key here is the Subtraction Property of Inequality. This property states that if you subtract the same number from both sides of an inequality, the inequality remains true. In simpler terms, it's like taking the same weight off both sides of a balanced scale – the balance (or, in this case, the inequality) is maintained. To get from $7x + 4 < 46$ to $7x < 42$, we subtract 4 from both sides of the inequality. This isolates the term with x on the left side, bringing us closer to our goal of solving for x. The subtraction property is a fundamental tool in solving inequalities because it allows us to simplify expressions without changing the solution set. It's like using a specific tool in a toolbox to loosen a particular bolt – it's the right technique for the job. Understanding and applying this property correctly is crucial for solving a wide range of inequalities. It's a building block upon which more complex problem-solving strategies are built. Remember, the Subtraction Property of Inequality is not just about subtracting numbers; it's about maintaining balance and preserving the truth of the inequality. It's a powerful concept that, when mastered, significantly enhances your ability to tackle mathematical challenges. It is also important to understand that this step is reversible. We can add 4 to both sides of $7x < 42$ and end up with the inequality in Step 2, this reversibility is a hallmark of algebraic manipulation that keeps the logical flow of the problem intact.

Step 4 $x < 6$

Unleashing the Division Property of Inequality The Final Stretch

We've reached the final step! In step 4, we go from $7x < 42$ to $x < 6$. This transformation involves the Division Property of Inequality. Similar to the Subtraction Property, the Division Property states that if you divide both sides of an inequality by the same positive number, the inequality remains true. The crucial word here is positive. When we divide (or multiply) both sides of an inequality by a negative number, we need to flip the direction of the inequality sign. This is because dividing by a negative number reverses the order of the numbers on the number line. However, in our case, we're dividing by 7, which is positive, so we don't need to worry about flipping the sign. Dividing both sides of $7x < 42$ by 7 isolates x on the left side, giving us the solution $x < 6$. This means any value of x less than 6 will satisfy the original inequality. We've successfully solved for x! The Division Property of Inequality is a powerful tool that allows us to simplify inequalities and isolate variables. It's like using a magnifying glass to zoom in on the solution – it brings clarity and precision to our answer. Mastering this property, along with the Subtraction Property, is essential for anyone looking to conquer the world of inequalities. Just remember the golden rule: divide by a positive number, and the sign stays the same; divide by a negative number, and flip the sign! This careful application of the Division Property culminates our efforts, leading us to the solution. It represents the final piece of the puzzle, revealing the range of values that satisfy the initial conditions.

Summarizing the Properties Used

Alright guys, let's recap the properties we used to solve the inequality $7x + 4 < 46$:

1. Given: This step simply states the original inequality.
2. Restating: We rewrote the inequality to ensure clarity.
3. Subtraction Property of Inequality: We subtracted 4 from both sides to get $7x < 42$.
4. Division Property of Inequality: We divided both sides by 7 to get $x < 6$.

By understanding and applying these properties, we can confidently solve a wide variety of inequalities. Remember, practice makes perfect, so keep working on those problems, and you'll become an inequality-solving pro in no time!

The Big Picture Why This Matters

Understanding the properties used in solving inequalities isn't just about getting the right answer; it's about developing a deeper understanding of mathematical principles. These properties are the building blocks of algebra, and mastering them will help you tackle more complex problems in the future. Think of it like learning the fundamentals of a sport – you need to know the basic rules and techniques before you can become a star player. In this case, the properties of inequality are the fundamental rules, and solving inequalities is the game. Moreover, the skills you gain from solving inequalities are transferable to other areas of life. The ability to think logically, break down problems into smaller steps, and apply the correct principles are valuable assets in any field. Whether you're planning a budget, designing a building, or writing a computer program, the problem-solving skills you develop in mathematics will serve you well. So, while solving inequalities might seem like an abstract exercise, it's actually a powerful way to develop critical thinking skills that will benefit you throughout your life. It's about more than just numbers and symbols; it's about building a foundation for success. Understanding the underlying logic and reasoning behind each step allows you to not only solve problems but also to adapt your strategies to new and unfamiliar situations. This adaptability is the hallmark of a true problem-solver, someone who can confidently approach challenges and find effective solutions. So, embrace the challenge of solving inequalities, and remember that each step you take is not just a move towards the answer but also a step towards becoming a more capable and confident thinker.

Practice Makes Perfect Sharpening Your Skills

To truly master the art of solving inequalities, practice is key. The more you work with these concepts, the more comfortable and confident you'll become. Start with simple inequalities and gradually move on to more complex ones. Try varying the numbers, signs, and operations to challenge yourself and deepen your understanding. Look for opportunities to apply these skills in real-world situations, such as comparing prices, calculating distances, or planning schedules. The more you can connect mathematical concepts to your everyday life, the more meaningful and memorable they will become. Don't be afraid to make mistakes – they're a natural part of the learning process. When you encounter an error, take the time to understand why it occurred and how you can avoid it in the future. Seek out resources such as textbooks, online tutorials, and practice problems to supplement your learning. Collaborate with classmates or friends to discuss challenging problems and share different approaches. Remember, learning mathematics is a journey, not a destination. Enjoy the process of discovery, and celebrate your progress along the way. Each problem you solve is a victory, and each concept you master is a step towards greater mathematical fluency. So, keep practicing, keep exploring, and keep pushing yourself to learn more. The world of mathematics is vast and fascinating, and the more you delve into it, the more rewards you will reap.

Conclusion

So there you have it, guys! We've successfully identified the properties used in each step of solving the inequality $7x + 4 < 46$. From stating the given information to applying the Subtraction and Division Properties of Inequality, we've broken down the process into manageable steps. Remember, understanding the “why” behind the “how” is crucial for mathematical success. By grasping the underlying principles, you'll be able to tackle any inequality that comes your way. Keep practicing, stay curious, and most importantly, have fun with math! You've got this!