Solving -5x + 4 < 0 A Step-by-Step Guide

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Hey guys! So, you're tackling inequalities and stumbled upon -5x + 4 < 0? No worries, I’ve got you covered! Inequalities might seem a little tricky at first, but once you break them down step-by-step, they become super manageable. This guide will walk you through the entire process, making sure you understand each move we make. Let’s dive in and conquer this inequality together!

Understanding Inequalities

Before we jump into solving our specific inequality, -5x + 4 < 0, let’s quickly recap what inequalities are all about. Think of inequalities as equations, but instead of an equals sign (=), we have symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These symbols tell us that the two sides of the expression aren't exactly equal, but rather one side is bigger or smaller than the other.

Now, when it comes to solving inequalities, our main goal is the same as solving equations: to isolate the variable (in this case, x) on one side of the inequality. The catch? We need to be a little careful with negative numbers. Multiplying or dividing both sides of an inequality by a negative number flips the direction of the inequality sign. Keep this in mind, and you'll be golden!

Why Inequalities Matter

Inequalities aren't just some abstract math concept; they pop up all over the place in real-world scenarios. Imagine you're trying to figure out how many hours you need to work to earn enough money for a new gadget, or you're calculating the possible range of values for a recipe ingredient. Inequalities are the tools that help us solve these kinds of problems. They allow us to deal with situations where we're looking for a range of solutions rather than a single, exact answer. So, mastering inequalities is not just about acing your math exams; it's about building a powerful problem-solving skill that you'll use in many different aspects of life. Trust me, once you get the hang of them, you’ll see opportunities to use them everywhere!

Step 1: Isolate the Term with 'x'

Okay, let's get our hands dirty with the actual problem: -5x + 4 < 0. The first thing we want to do is isolate the term that contains our variable, which is -5x. To do this, we need to get rid of that + 4 hanging out on the left side. Remember, our goal is to get -5x all by itself on one side of the inequality.

So, how do we do that? We use the same principle we use when solving regular equations: we perform the inverse operation. In this case, we have + 4, so we need to subtract 4 from both sides of the inequality. This keeps the inequality balanced, just like with equations. Think of it like a scale – if you take something off one side, you need to take the same thing off the other side to keep it even.

Let's do it: -5x + 4 - 4 < 0 - 4. When we simplify this, the + 4 and - 4 on the left side cancel each other out, leaving us with -5x < -4. Great! We've successfully isolated the term with x. We're one step closer to solving the inequality. This step is super important because it sets us up for the final move, where we actually solve for x. So, make sure you’re comfortable with this idea of isolating the variable term – it's a fundamental skill for solving all sorts of inequalities.

Step 2: Solve for 'x'

Alright, we've got -5x < -4. Now comes the crucial step: solving for x. Notice that x is currently being multiplied by -5. To isolate x, we need to get rid of that -5. Just like in the previous step, we'll use the inverse operation. Since -5 is multiplying x, we need to divide both sides of the inequality by -5.

But hold on! Remember what we talked about earlier? There's a golden rule when dealing with inequalities and negative numbers: when you multiply or divide both sides of an inequality by a negative number, you have to flip the direction of the inequality sign. This is super important, so don't forget it!

So, let's divide both sides by -5 and flip the sign: (-5x) / (-5) > (-4) / (-5). Notice how the < sign has changed to a > sign. This is the key to getting the correct solution. Now, let's simplify. On the left side, -5 divided by -5 is just 1, so we're left with x. On the right side, -4 divided by -5 is 4/5. So, our inequality now reads x > 4/5.

What does this mean? It means that any value of x that is greater than 4/5 will satisfy the original inequality, -5x + 4 < 0. We've successfully solved for x! This step is where a lot of people can make mistakes, especially with the negative sign rule, so double-check your work and make sure you've flipped the inequality sign when necessary.

Step 3: Expressing the Solution

We've found that x > 4/5, but how do we express this solution in a way that's crystal clear? There are a couple of ways to do this: using inequality notation, which we already have, and using interval notation. Let's break down both.

Inequality Notation

Inequality notation is what we've been using all along: x > 4/5. This is a straightforward way to say that x can be any number greater than 4/5. It's simple and to the point. However, sometimes it's helpful to express the solution in a slightly different way, especially when we're dealing with more complex inequalities or when we need to graph the solution.

Interval Notation

Interval notation is a way of expressing a range of numbers using parentheses and brackets. It might look a little intimidating at first, but it's actually quite handy once you get the hang of it. Here's the basic idea:

  • Parentheses () indicate that the endpoint is not included in the interval.
  • Brackets [] indicate that the endpoint is included in the interval.
  • Infinity ∞ and negative infinity -∞ are used to represent unbounded intervals (intervals that go on forever). We always use parentheses with infinity because we can't actually