Solving Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of inequalities and tackling the problem: -3x + 6 + 12(x + 1) < x + 6. Don't worry, it might seem a bit intimidating at first, but trust me, with a few simple steps, we'll break it down and understand how to solve it. Inequalities are like equations, but instead of an equal sign (=), we have symbols like less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). Solving them is all about isolating the variable (in this case, 'x') to find the range of values that make the inequality true. So, grab your pencils and let's get started! We will explore each step in detail so you can grasp the concepts and apply them to similar problems. This article provides a comprehensive guide for anyone looking to master the art of solving inequalities. Whether you're a student struggling with algebra or just someone curious about math, this is the perfect starting point to understanding and solving inequalities. Let's start with the basics.

Simplifying the Inequality: The First Steps

Alright guys, the first step in solving this inequality is to simplify it. This means getting rid of those pesky parentheses and combining like terms. Let's start by looking at the left side of the inequality: -3x + 6 + 12(x + 1). Remember the order of operations (PEMDAS/BODMAS)? We have to deal with the parentheses first. So, we'll distribute the 12 to both terms inside the parentheses. That means multiplying 12 by 'x' and then multiplying 12 by 1. This gives us: 12 * x = 12x and 12 * 1 = 12. So, our expression becomes: -3x + 6 + 12x + 12 < x + 6. See? Not so scary, right? Now, we can combine the like terms on the left side. We have '-3x' and '+12x', which combine to give us '9x'. We also have the constants '+6' and '+12', which add up to 18. Therefore, our simplified inequality now looks like: 9x + 18 < x + 6. We're making progress! We are effectively reducing the complexity of the equation, making it easier to solve. The concept of simplifying is crucial to solving not only this type of inequality, but also many other algebraic equations and problems. This is a fundamental concept in mathematics. Remember, the goal is always to isolate the variable, and simplifying is the first and most important step to achieving that goal.

Isolating the Variable: Getting 'x' Alone

Okay, team, now comes the part where we want to isolate 'x'. To do this, we need to get all the 'x' terms on one side of the inequality and all the constant terms on the other side. Let's start by subtracting 'x' from both sides of the inequality. This eliminates the 'x' on the right side. Our equation 9x + 18 < x + 6 now becomes: 9x - x + 18 < x - x + 6. This simplifies to: 8x + 18 < 6. Next, we need to get rid of the '+18' on the left side. We do this by subtracting 18 from both sides. This keeps the inequality balanced. So, we get: 8x + 18 - 18 < 6 - 18, which simplifies to: 8x < -12. See? We're getting closer! The goal is to keep the inequality balanced at all times; whatever operation we perform on one side, we must perform on the other side as well. This is because the values on both sides of the inequality must remain in the same order. By performing the same operations on both sides, we make sure that we're keeping the relationships between the values constant and valid. When we isolate the variable, we find the range of x values that satisfy the original inequality. In the context of solving inequalities, the objective is to determine a range of possible values for the unknown variable.

Solving for 'x' and Understanding the Solution

Almost there, folks! The final step is to solve for 'x'. We have 8x < -12. To isolate 'x', we need to divide both sides of the inequality by 8. This gives us: 8x / 8 < -12 / 8. Which simplifies to: x < -1.5. And that's it! We've solved the inequality. The solution, x < -1.5, means that any value of 'x' that is less than -1.5 will make the original inequality true. So, if you plug in any number smaller than -1.5 (like -2, -3, or even -100), you'll find that the inequality holds true. If we had, for instance, divided by a negative number in the process of solving, we would have had to flip the inequality sign. But in this case, we have our answer without that happening. It's a critical detail to keep in mind when solving inequalities. Understanding the solution is extremely important. We found that x must be less than -1.5. This means that any value of x, that, when substituted in the original equation, results in a true statement. So you can pick a number that is less than -1.5, such as -2, and substitute it into the original equation to see that both sides are balanced and the inequality is true.

Graphing the Solution: Visualizing the Range

To better understand the solution, it's helpful to visualize it on a number line. Draw a number line and mark the point -1.5. Since our solution is x < -1.5, we'll use an open circle (or parenthesis) at -1.5 to indicate that -1.5 itself is not included in the solution. Then, we'll draw an arrow going to the left from -1.5. This arrow represents all the numbers that are less than -1.5. This kind of visualization helps us grasp the concept of the solution. If the inequality was x ≤ -1.5 (less than or equal to), we would use a closed circle (or bracket) at -1.5 to show that -1.5 is included in the solution. Graphing the solution gives you a clear picture of the range of values that satisfy the inequality. It provides an immediate visual representation and aids in understanding the extent of the solution. This visual is often useful in various fields, especially where modeling data is involved. Understanding how to graph inequalities is a beneficial skill.

Checking Your Answer: Always a Good Idea

Always a good idea to check if your answer is correct, right? So let's pick a number that is less than -1.5 and plug it into the original inequality to make sure it works. Let's pick -2. Our original inequality was: -3x + 6 + 12(x + 1) < x + 6. Substituting -2 for 'x', we get: -3(-2) + 6 + 12(-2 + 1) < -2 + 6. Simplifying this, we get: 6 + 6 + 12(-1) < 4. Which becomes: 6 + 6 - 12 < 4. And then: 0 < 4. This statement is true! Since our final statement is true, we know our solution, x < -1.5, is correct. That's a great feeling, isn't it? Checking your work is an essential practice in mathematics. It helps to catch any mistakes you may have made along the way and reinforces your understanding of the concepts. It can also help you avoid the frustration of getting the wrong answer on a test or in a real-world situation. It provides a means to confirm that the solution aligns with the constraints of the original problem.

Real-World Applications of Inequalities

Inequalities aren't just abstract math concepts; they have real-world applications! They're used in various fields, from business and economics to science and engineering. For example, inequalities are used to determine profit margins. They can show how much a product needs to be sold for to make a profit. They help in setting minimum and maximum values. Think about it: when setting a budget, you might use inequalities to ensure your expenses stay below a certain limit. In economics, inequalities are used to model supply and demand relationships. Engineers use inequalities to define the safe operating ranges of machinery. Even in everyday life, you might use an inequality when you want to make sure you have enough money to buy groceries. Because these real-world scenarios are complex, a good understanding of inequalities is essential to solve the more complicated problems.

Conclusion: Mastering the Art of Inequalities

And there you have it, folks! We've successfully solved the inequality -3x + 6 + 12(x + 1) < x + 6. We broke it down step by step, simplified, isolated the variable, and found our solution: x < -1.5. Remember the key steps: simplify, isolate, and solve. Practice is key, so try solving some more inequalities on your own. You'll get better with each problem you solve. Make sure to understand each step. With practice, you'll become a pro at solving inequalities. Keep practicing, and you'll find that inequalities become second nature. Keep asking questions and exploring, and you'll be well on your way to mathematical mastery! Keep practicing, and don't be afraid to ask for help when you need it. Math can be tricky, but it's also rewarding when you finally get that "aha!" moment. Congrats and keep up the great work! Now go out there and conquer those inequalities!