Solving Inequalities: Properties Used In 7x+4 < 46

Hey guys! Today, we're diving into the world of inequalities and taking a look at how we can solve them step-by-step. We'll focus on the specific inequality 7x + 4 < 46 and break down each step, identifying the property that allows us to move forward. Understanding these properties is super important for mastering algebra, so let's get started!

1. The Given Inequality: 7x + 4 < 46

So, when we start with an inequality like 7x + 4 < 46, the first thing we acknowledge is that this is our given statement. It’s the foundation upon which we'll build our solution. Think of it as the starting point of a journey – we need a map (our understanding of mathematical properties) to reach our destination (the solution). In this initial step, we're not actually doing anything except recognizing what we've been given. This might seem obvious, but explicitly stating “Given” is a crucial step in formal mathematical problem-solving, especially when you're learning. It helps to establish a clear and logical progression of your work. By acknowledging the given information, we set the stage for applying various properties of inequalities to isolate the variable x and determine the range of values that satisfy the inequality. So, remember, even the simple act of stating the given is a property in itself – the property of acknowledging the initial condition. We need this starting point to apply the properties of equality and inequality that will lead us to the solution. The beauty of mathematics lies in its precision and logical structure, and explicitly stating the given is the first brick in building that structure. This foundation ensures that each subsequent step is justified and flows logically from the previous one, ultimately leading us to a correct and well-understood solution.

2. Subtracting 4 from Both Sides: 7x < 42

Now, let's talk about how we get from 7x + 4 < 46 to 7x < 42. This step involves a fundamental property called the Subtraction Property of Inequality. This property is a game-changer because it allows us to manipulate inequalities while keeping them balanced. Imagine an inequality as a seesaw – to maintain balance, whatever you do on one side, you must do on the other. The Subtraction Property states exactly this: if you subtract the same number from both sides of an inequality, the inequality remains true. In our case, we're subtracting 4 from both sides. Why 4? Because it's the key to isolating the term with x (which is 7x). By subtracting 4, we effectively “undo” the addition of 4 on the left side of the inequality. This brings us closer to our goal of getting x by itself. So, when we subtract 4 from both sides, we get: 7x + 4 - 4 < 46 - 4, which simplifies to 7x < 42. This might seem like a small step, but it's a crucial one. We've successfully eliminated the constant term on the left side, making our inequality simpler and easier to work with. The Subtraction Property of Inequality is a powerful tool in our mathematical arsenal, and it’s essential for solving a wide range of inequalities. Remember, it's all about maintaining balance and ensuring that the relationship between the two sides of the inequality remains consistent throughout the solution process. Understanding and applying this property correctly is a major step towards mastering inequality problems.

3. Dividing Both Sides by 7: x < 6

Okay, we've reached the final step in solving our inequality 7x < 42. To isolate x completely, we need to get rid of the coefficient 7 that's multiplying it. This is where the Division Property of Inequality comes into play. Similar to the Subtraction Property, the Division Property helps us maintain the balance of the inequality. It states that if you divide both sides of an inequality by the same positive number, the inequality remains true. Notice that I emphasized positive – this is a crucial detail! When dividing (or multiplying) by a negative number, you need to flip the inequality sign, but we'll save that for another example. In our case, we're dividing both sides by 7, which is a positive number. So, we can confidently apply the Division Property without worrying about flipping the sign. When we divide both sides of 7x < 42 by 7, we get: (7x) / 7 < 42 / 7. This simplifies beautifully to x < 6. And there we have it! We've successfully solved the inequality. This result tells us that any value of x that is less than 6 will satisfy the original inequality 7x + 4 < 46. The Division Property of Inequality is an indispensable tool for solving inequalities, and it's important to remember the positive number caveat. Mastering this property, along with the Subtraction Property, will equip you to tackle a wide array of inequality problems. So, remember to divide carefully, keep an eye on the sign, and you'll be solving inequalities like a pro in no time!

Conclusion

So, guys, we've successfully navigated the steps to solve the inequality 7x + 4 < 46, identifying the properties used along the way. We started with the Given inequality, then used the Subtraction Property of Inequality to simplify it, and finally applied the Division Property of Inequality to isolate x. The solution, x < 6, tells us that any number less than 6 will make the original inequality true. Remember, understanding these properties is key to solving not just this inequality, but many others you'll encounter in your math journey. Keep practicing, and you'll become a pro at solving inequalities in no time!