Solving Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of inequalities. They might seem a bit tricky at first, but trust me, with a little practice, you'll be solving them like a pro. Today, we're going to break down the inequality $6x geq 3+4(2x-1)$ and figure out its correct representations. We'll go through the process step-by-step, making sure you understand every move. Ready? Let's get started!

Understanding the Basics of Inequalities

First things first, what exactly is an inequality? Well, it's a mathematical statement that compares two expressions using symbols like greater than $(>)$, less than $(<)$, greater than or equal to $(\geq)$, or less than or equal to $(\leq)$. Unlike equations, which have a single solution, inequalities often have a range of solutions. Think of it like this: an equation is like finding the exact weight of an object, while an inequality is like saying the weight is at least this much. The key to solving inequalities is to isolate the variable, just like you would in an equation. However, there's a crucial rule to remember: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is the golden rule, and it's essential to get the correct answer. For example, if we have $-2x > 4$, dividing both sides by $-2$ gives us $x < -2$ (note the sign flip!). Now, let's look at the given inequality and see how we can solve it step by step. This foundational understanding is the bedrock upon which our problem-solving skills will be built. Grasping these core concepts sets the stage for successfully tackling more complex problems. It's like learning the alphabet before you start writing a novel. The better you understand these fundamentals, the easier it will be to navigate the intricacies of the problem. Remember, practice makes perfect. The more you work with inequalities, the more comfortable and confident you'll become.

Breaking Down the Inequality: $6x

geq 3+4(2x-1)$

Our initial inequality is $6x geq 3+4(2x-1)$. The first thing we want to do is simplify the right side of the inequality. We'll use the distributive property to multiply the $4$ by both terms inside the parentheses: $4(2x-1)$. That means we multiply $4$ by $2x$ and $4$ by $-1$. This gives us $8x - 4$. So now, our inequality becomes $6x geq 3 + 8x - 4$. See, not so bad, right? We're just applying basic algebra rules. Think of each step as a mini-puzzle piece that, when combined, reveals the full picture. The distributive property is one of the most fundamental tools in algebra, and it's used to remove parentheses. Make sure you understand how it works because you'll encounter it frequently. Remember, the goal is always to simplify and isolate the variable. The careful application of these steps ensures we maintain the correct relationship between the two sides of the inequality. Pay attention to the signs and keep everything organized. It helps to write each step clearly to avoid making careless errors. Mistakes can happen, but they become less likely when you follow a methodical approach.

Combining Like Terms and Isolating the Variable

Now, let's simplify further. On the right side, we have $3$ and $-4$. These are like terms, so we can combine them: $3 - 4 = -1$. Our inequality now looks like this: $6x geq 8x - 1$. The next step is to get all the $x$ terms on one side of the inequality. We can do this by subtracting $8x$ from both sides. This gives us $6x - 8x geq -1$. This step is designed to bring the variable terms together. This simplifies the equation so that the variable terms are grouped. Remember, anything you do to one side of an inequality, you must do to the other side to keep it balanced. This ensures that the relationship between the two sides remains true. It's like balancing a scale; if you add or remove weight from one side, you must do the same to the other side to maintain equilibrium. Always double-check your arithmetic in each step. Even a small error can lead to a wrong answer. A methodical approach, including careful attention to detail, is critical here. These small steps will ensure a clear path to solving the inequality. The concept of isolating the variable is fundamental to algebra. The more familiar you become with this process, the faster and more accurately you'll be able to solve inequalities. This systematic process avoids confusion and sets a solid foundation for future concepts. The more practice you get, the easier this process will become.

Simplifying and Finding the Solution

Now, let's simplify $6x - 8x$. This gives us $-2x$. So, we have $-2x geq -1$. To isolate $x$, we need to divide both sides by $-2$. Remember our golden rule? Since we're dividing by a negative number, we must flip the inequality sign. So, $-2x geq -1$ becomes $x leq \frac{1}{2}$. That's it! We've solved the inequality. The solution is $x leq \frac{1}{2}$. This means that any value of $x$ that is less than or equal to $\frac{1}{2}$ will satisfy the original inequality. Double-checking each step makes sure your work is correct. Dividing both sides by a negative number and flipping the inequality sign is a very important step and the source of many errors. Make sure you understand why we do this. The result $x leq \frac{1}{2}$ represents all the possible values of $x$ that are solutions to the inequality. This understanding helps to ensure you can apply the same techniques to different problems. The careful steps that you take ensure a complete and correct solution. Recognizing the different forms of the solution is important. Practicing these types of problems will boost your math abilities and understanding. The confidence gained from correctly solving an inequality is priceless. This systematic approach not only solves the problem but also builds a strong foundation for future mathematical endeavors. Remember, consistent practice will solidify your skills and provide a solid foundation for more complex mathematical concepts.

Checking the Answer Choices

Now let's examine the answer choices. We are given two options, $6x geq 3 + 8x - 4$ and $1 geq 2x$. We've already shown how to simplify the original inequality to $6x geq 3 + 8x - 4$. Since we used the distributive property in the initial step, this statement must be correct. Next, if we simplify $6x geq 3 + 8x - 4$, we combine like terms to get $6x geq 8x - 1$. Then, subtract $8x$ from both sides: $-2x geq -1$. Divide both sides by $-2$ and remember to flip the sign, so we get $x leq \frac{1}{2}$. Multiplying both sides by 2 is $2x \leq 1$, which is the same as the second option, $1 geq 2x$. We can check the solution using graphing calculators and online tools, too. These methods not only confirm your answers but also provide additional practice. Keep in mind that understanding the steps is far more valuable than simply arriving at an answer. Being able to explain each step and justify your reasoning is the mark of true understanding. You can also substitute values to test your solution. This will help you to verify your answer. This step is about verifying the solution.

Conclusion: Mastering Inequalities

We've covered the basics of solving inequalities, simplified $6x geq 3+4(2x-1)$, combined like terms, isolated the variable, and correctly identified the answer choices. Remember the golden rule: flip the inequality sign when multiplying or dividing by a negative number! Keep practicing, and you'll become a pro at solving inequalities. Always take things step by step, and don't be afraid to ask for help if you need it. Math is a journey, not a destination. Each problem you solve brings you closer to mastery. Good luck, and happy solving!