Solving Inequalities A Step-by-Step Guide To 2(-4x + 2) ≥ -5x - 10

Hey there, math enthusiasts! Today, we're diving deep into the world of inequalities, specifically tackling the problem 2(-4x + 2) ≥ -5x - 10. Inequalities might seem a bit intimidating at first, but trust me, with a systematic approach, they become a piece of cake. We'll break down each step, explain the logic behind it, and make sure you're confident in solving similar problems. So, grab your pencils, and let's get started!

1. Understanding the Foundation: What are Inequalities?

Before we jump into solving the problem, let's quickly recap what inequalities are all about. Unlike equations that have a single solution, inequalities represent a range of possible solutions. Think of it as a spectrum rather than a specific point. Inequalities use symbols like "greater than" (>), "less than" (<), "greater than or equal to" (≥), and "less than or equal to" (≤) to express these relationships. In our case, we're dealing with "greater than or equal to" (≥), which means our solution will include all values of x that are either greater than or equal to a certain number. Understanding this fundamental concept is crucial before attempting any inequality problems.

Now, let's relate this to our specific problem: 2(-4x + 2) ≥ -5x - 10. This inequality states that the expression on the left side, 2(-4x + 2), must be greater than or equal to the expression on the right side, -5x - 10. Our goal is to isolate 'x' and find the range of values that satisfy this condition. We'll achieve this by performing algebraic operations on both sides of the inequality, making sure to maintain the balance and direction of the inequality. Remember, just like with equations, whatever we do to one side, we must also do to the other. However, there's one crucial difference when dealing with inequalities: multiplying or dividing by a negative number flips the direction of the inequality sign. Keep this in mind as we proceed!

2. The First Step: Distribute and Conquer

The first order of business is to simplify the inequality by getting rid of the parentheses. We achieve this by applying the distributive property, which essentially means multiplying the term outside the parentheses by each term inside. In our case, we need to distribute the '2' in 2(-4x + 2). So, 2 multiplied by -4x gives us -8x, and 2 multiplied by +2 gives us +4. Therefore, 2(-4x + 2) becomes -8x + 4. This transforms our original inequality into a more manageable form: -8x + 4 ≥ -5x - 10. Mastering the distributive property is essential in algebra, and it's a powerful tool in simplifying complex expressions.

By applying the distributive property, we've successfully removed the parentheses and paved the way for further simplification. Now, our inequality looks cleaner and more approachable. The next step involves gathering like terms, which will bring us closer to isolating 'x'. Think of it as organizing your toolbox before tackling a project – having everything in its place makes the job much smoother. We're essentially grouping the 'x' terms on one side and the constant terms on the other side. This will allow us to combine them and simplify the inequality even further. So, let's move on to the next step and see how we can bring these terms together!

3. Gathering the Troops: Combining Like Terms

Now that we have -8x + 4 ≥ -5x - 10, it's time to gather our like terms. Our mission here is to bring all the 'x' terms to one side of the inequality and all the constant terms to the other side. This makes the equation simpler and easier to solve. To do this, we'll use the addition property of inequality, which states that adding the same value to both sides of an inequality preserves the inequality. Let’s start by adding 5x to both sides of the inequality. This will eliminate the -5x term on the right side and bring the 'x' terms closer together. Adding 5x to both sides gives us: -8x + 4 + 5x ≥ -5x - 10 + 5x. Simplifying this, we get -3x + 4 ≥ -10. Remember, our goal is to isolate 'x', and this step brings us closer to achieving that.

Having consolidated the 'x' terms, we now shift our focus to the constant terms. We have +4 on the left side, which we need to move to the right side. Again, we'll use the addition property of inequality, but this time we'll subtract 4 from both sides. This will eliminate the +4 on the left side and group the constant terms on the right. Subtracting 4 from both sides gives us: -3x + 4 - 4 ≥ -10 - 4. Simplifying this, we arrive at -3x ≥ -14. Now, our inequality is in a much simpler form, with the 'x' term isolated on the left and the constant term on the right. We're almost there! Just one more step, and we'll have our solution. This next step involves dealing with the coefficient of 'x', which will finally reveal the range of values that satisfy our inequality. So, let's proceed and conquer this final hurdle!

4. The Final Showdown: Isolating 'x'

We've reached the crucial step of isolating 'x' in the inequality -3x ≥ -14. To do this, we need to get rid of the coefficient -3 that's multiplying 'x'. This is where the division property of inequality comes into play. We'll divide both sides of the inequality by -3. However, remember the golden rule: when you multiply or divide an inequality by a negative number, you must flip the direction of the inequality sign. This is a critical step that students often overlook, and it can lead to incorrect solutions. So, let's be extra careful here. Dividing both sides by -3 gives us: (-3x) / -3 ≤ (-14) / -3. Notice how the "greater than or equal to" (≥) sign has flipped to "less than or equal to" (≤). Simplifying this, we get x ≤ 14/3. This is our solution!

We've successfully isolated 'x' and found the range of values that satisfy the original inequality. The solution x ≤ 14/3 tells us that any value of 'x' that is less than or equal to 14/3 will make the inequality true. To fully understand and communicate our solution, we need to express it in interval notation. This is a standard way of representing a range of values, and it provides a concise and clear way to convey the solution set. So, let's take a look at what interval notation is and how we can use it to represent our solution.

5. Expressing the Solution: Interval Notation

Now that we've solved for x and found x ≤ 14/3, let's express this solution using interval notation. Interval notation is a way of writing sets of numbers using intervals. It's a concise and clear method to represent a range of values, especially when dealing with inequalities. The basic idea is to use brackets and parentheses to indicate whether the endpoints of the interval are included or excluded, respectively. A square bracket [ or ] indicates that the endpoint is included in the interval, while a parenthesis ( or ) indicates that the endpoint is not included. Infinity (∞) and negative infinity (-∞) are always enclosed in parentheses because they are not actual numbers and cannot be included as endpoints.

In our case, x ≤ 14/3 means that x can be any value less than or equal to 14/3. This includes 14/3 itself and extends all the way to negative infinity. In interval notation, this is represented as (-∞, 14/3]. The parenthesis next to -∞ indicates that negative infinity is not included, and the square bracket next to 14/3 indicates that 14/3 is included in the solution set. This interval notation provides a complete and unambiguous representation of our solution. Anyone familiar with interval notation can quickly understand that the solution includes all numbers from negative infinity up to and including 14/3. So, we've not only solved the inequality but also expressed the solution in a standard and widely accepted mathematical notation. This completes our journey! We've successfully tackled the inequality, step by step, and arrived at a clear and concise solution.

6. In Conclusion: The Beauty of Systematic Problem-Solving

We've successfully navigated the inequality 2(-4x + 2) ≥ -5x - 10, and hopefully, you've gained a deeper understanding of how to approach these problems. The key takeaway here is the power of a systematic approach. By breaking down a complex problem into smaller, manageable steps, we can conquer even the most daunting challenges. We started by understanding the fundamentals of inequalities, then meticulously worked through the steps of distribution, combining like terms, isolating the variable, and finally, expressing the solution in interval notation. Each step built upon the previous one, leading us to a clear and accurate solution.

Remember, math isn't just about memorizing formulas; it's about developing a logical and structured way of thinking. Inequalities, like many mathematical concepts, require a blend of understanding the underlying principles and applying the correct techniques. So, keep practicing, keep exploring, and don't be afraid to tackle new challenges. The more you engage with these concepts, the more confident and proficient you'll become. And who knows, maybe you'll even start to enjoy the beauty and elegance of mathematics! Happy problem-solving, guys!