Solving -1/4(3x - 1/2) > 0 Finding The Solution Set

Hey guys! Let's dive into solving inequalities, specifically the one you asked about: -1/4(3x - 1/2) > 0. Inequalities might seem a bit tricky at first, but with a step-by-step approach, we can totally conquer them. This article will guide you through the process, making sure you understand each move we make. We'll break down the problem, explain the concepts, and ensure you're confident in finding the solution set. Think of it as a friendly walkthrough rather than a daunting math problem. So, let's get started and make those inequalities our friends!

Understanding Inequalities

Before we jump into solving the specific inequality, let's take a moment to really understand what inequalities are all about. In the world of math, an inequality is like a comparison tool. Instead of saying things are exactly equal (which is what the "=" sign does), inequalities help us express when one thing is greater than, less than, greater than or equal to, or less than or equal to another thing. Think of it like comparing the heights of your friends – maybe you're taller than one friend, shorter than another, or the same height as someone else. Inequalities do the same thing, but with numbers and expressions.

The symbols we use to represent these relationships are super important. We've got '> 'for greater than, '<' for less than, '≥' for greater than or equal to, and '≤' for less than or equal to. Getting comfy with these symbols is the first step to mastering inequalities. They're the language we use to describe these comparisons mathematically. For instance, 'x > 5' means that 'x' is any number bigger than 5, but not 5 itself. If we wanted to include 5, we'd use 'x ≥ 5'.

Now, how do inequalities show up in real life? Well, they're everywhere! Imagine you're planning a party and you have a budget. You might use an inequality to figure out how many guests you can invite without spending more than your budget allows. Or, think about speed limits on the road – they're expressed as inequalities, telling you the maximum speed you can drive. Inequalities also play a big role in science, engineering, and economics, helping us model and solve problems where things aren't always exact but fall within a certain range. So, understanding inequalities isn't just about acing a math test; it's about building a skill that's useful in tons of different situations. The fun is about to begin, so let's dive in!

Step-by-Step Solution for -1/4(3x - 1/2) > 0

Alright, let's get into the heart of the matter and tackle the inequality -1/4(3x - 1/2) > 0. Don't worry, we'll break it down into easy-to-follow steps. Think of it like following a recipe – each step builds on the previous one, leading us to the final delicious solution. Solving inequalities is super similar to solving equations, but there's one key difference we'll highlight along the way. So, let's roll up our sleeves and get started!

Step 1: Distribute the -1/4. Our first move is to get rid of those parentheses. We do this by distributing the -1/4 across the terms inside. This means we multiply -1/4 by both 3x and -1/2. So, -1/4 multiplied by 3x gives us -3/4x, and -1/4 multiplied by -1/2 (a negative times a negative!) gives us +1/8. Our inequality now looks like this: -3/4x + 1/8 > 0.

Step 2: Isolate the x term. Next up, we want to get the term with 'x' on its own on one side of the inequality. To do this, we'll subtract 1/8 from both sides. This keeps the inequality balanced, just like keeping a scale balanced. Subtracting 1/8 from both sides gives us: -3/4x > -1/8.

Step 3: Solve for x (and the crucial twist!). Now comes the most important part! We need to get 'x' completely alone. To do this, we'll divide both sides by -3/4. Here's the super important thing to remember: when you multiply or divide both sides of an inequality by a negative number, you have to flip the inequality sign. This is the key difference between solving inequalities and solving equations. So, when we divide both sides by -3/4, our '>' sign becomes a '<' sign. Doing the division, we get x < (-1/8) / (-3/4). To divide by a fraction, we multiply by its reciprocal, so we have x < (-1/8) * (-4/3). Multiplying those fractions gives us x < 4/24, which simplifies to x < 1/6.

Step 4: Express the solution set. We've found that x must be less than 1/6. How do we write that as a solution set? Well, it's all the numbers that fit that description. In interval notation, we write this as (-∞, 1/6). This means all numbers from negative infinity up to, but not including, 1/6. On a number line, we'd represent this with an open circle at 1/6 and an arrow extending to the left, showing all the numbers less than 1/6 are solutions. And that's it! We've solved the inequality and expressed the solution set. Pretty awesome, right?

Representing the Solution Set

Now that we've found our solution, x < 1/6, it's super important to know how to represent it in different ways. This helps us really understand what the solution means and communicate it effectively. There are two main ways we'll focus on: interval notation and graphical representation on a number line.

Interval Notation: Interval notation is like a mathematical shorthand for describing a range of numbers. It uses parentheses and brackets to show whether the endpoints of the range are included or not. In our case, x < 1/6 means all numbers less than 1/6, but not 1/6 itself. So, we use a parenthesis next to 1/6. Since we're going all the way to negative infinity, we use (-∞. Infinity always gets a parenthesis because we can't actually reach infinity – it's a concept, not a specific number. Putting it all together, our solution in interval notation is (-∞, 1/6). Remember, the parenthesis tells us that 1/6 is not included in the solution set. If our solution had been x ≤ 1/6, we would have used a bracket, like this: (-∞, 1/6], to show that 1/6 is included.

Graphical Representation: Visualizing the solution on a number line is another powerful way to understand it. Draw a straight line and mark the key number in our solution, which is 1/6. Since x is less than 1/6, we draw an open circle at 1/6. This open circle tells us that 1/6 is not part of the solution. Then, we draw an arrow extending to the left from 1/6. This arrow represents all the numbers less than 1/6, stretching all the way to negative infinity. If our solution had included 1/6 (x ≤ 1/6), we would have filled in the circle to show that 1/6 is part of the solution. The number line gives us a clear picture of all the values that satisfy our inequality. It's like seeing the solution laid out in front of us! Understanding both interval notation and graphical representation gives you a solid grasp of what the solution means and how to express it.

Common Mistakes and How to Avoid Them

When it comes to solving inequalities, there are a few common pitfalls that students often stumble into. But don't worry, we're going to shine a light on these mistakes so you can steer clear of them. Knowing what to watch out for is half the battle! Let's look at some frequent errors and how to make sure you don't fall into the same traps.

Forgetting to Flip the Inequality Sign: This is the big one! As we discussed earlier, whenever you multiply or divide both sides of an inequality by a negative number, you absolutely must flip the inequality sign. If you don't, you'll end up with the wrong solution. So, always double-check your steps, and if you're multiplying or dividing by a negative, make that sign flip! It's like a little dance move for inequalities – negative multiplication or division? Flip that sign!

Incorrectly Distributing Negatives: When distributing a negative number, it's crucial to apply it to every term inside the parentheses. It's easy to forget the negative sign on one of the terms, especially if there are multiple terms inside. A good way to avoid this is to take your time and write out each step clearly. Double-check that you've multiplied the negative by every single term. Think of it like giving everyone a fair share – each term gets the negative treatment!

Mixing Up Inequality Symbols: It's easy to mix up '> 'and '<', or '≥' and '≤'. A helpful tip is to think of the inequality symbol as an alligator's mouth – it always wants to