Solving Inequalities: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of inequalities. Specifically, we'll tackle two problems: 2x + 5 > 6x + 4 and -4x + 9 < x - 1. Don't worry, it's not as scary as it looks. We'll break it down step by step, making sure you grasp every concept. Think of inequalities as similar to equations, but instead of an equals sign (=), we have symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Our goal is to isolate the variable, 'x', just like we would in a regular equation. Let's get started, shall we?

Solving the Inequality: 2x + 5 > 6x + 4

Alright, let's roll up our sleeves and tackle the first inequality: 2x + 5 > 6x + 4. Our mission? To find the values of 'x' that make this statement true. Here's how we'll do it, one step at a time. First things first, we want to get all the 'x' terms on one side of the inequality. To do this, let's subtract 6x from both sides. This gives us: 2x - 6x + 5 > 6x - 6x + 4. Simplify that bad boy, and we have -4x + 5 > 4. See? We're already making progress. Now, our next move is to isolate the 'x' term. We can do this by subtracting 5 from both sides of the inequality. This gives us: -4x + 5 - 5 > 4 - 5. Simplify again, and we get -4x > -1. Almost there! Now comes the slightly tricky part. We need to solve for 'x', but we have a negative coefficient (-4) in front of it. To get 'x' by itself, we'll divide both sides by -4. Remember this crucial rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. So, our inequality -4x > -1 becomes -4x / -4 < -1 / -4. This gives us x < 1/4. And there you have it! Our solution is x < 1/4. This means any value of 'x' that is less than 1/4 will make the original inequality true. We did it, guys!

To make sure you understand, let's take an example and plug it into our original inequality. If x is equal to 0, it means 2(0) + 5 > 6(0) + 4 which is equal to 5 > 4. Our expression is true. And it has to be, because we have x < 1/4. Let's try x equal to 1. In this case, 2(1) + 5 > 6(1) + 4, which is equal to 7 > 10. That's not right! So our formula and our explanation make total sense.

Solving the Inequality: -4x + 9 < x - 1

Okay, let's switch gears and tackle our second inequality: -4x + 9 < x - 1. Same drill – our goal is to isolate 'x' and find its possible values. Let's start by getting all the 'x' terms on one side. We can do this by adding 4x to both sides: -4x + 4x + 9 < x + 4x - 1. This simplifies to 9 < 5x - 1. Awesome, the x's are starting to group together. Next, we want to get rid of that pesky -1. Let's add 1 to both sides: 9 + 1 < 5x - 1 + 1. Simplifying gives us 10 < 5x. Now, we want to isolate 'x'. We can do this by dividing both sides by 5: 10 / 5 < 5x / 5. Which gives us 2 < x. We can also write this as x > 2. So, our solution is x > 2. Any value of 'x' that is greater than 2 will satisfy this inequality. And with this, we completed our second inequality!

As you can see, the process is quite similar to solving equations. The main difference? That flipping of the inequality sign when multiplying or dividing by a negative number. That's the one thing to keep in your mind.

Let's test our formula again. If x = 3, then -4(3) + 9 < 3 - 1, which means -3 < 2. And it's true! Our inequality shows that x must be greater than 2. Let's try it with x = 1. -4(1) + 9 < 1 - 1, which means 5 < 0. That's not right. And it has to be this way, or we did something wrong in our formula. But it seems we're good to go!

Visualizing Solutions: Number Lines

Okay, guys, now that we've found our solutions, let's visualize them using number lines. This is a great way to understand the range of values that satisfy our inequalities. For the first inequality, x < 1/4, draw a number line. Mark 1/4 on the number line. Since x is less than 1/4 (and not equal to it), we'll use an open circle at 1/4. Then, shade the number line to the left of 1/4, representing all the values of 'x' that are less than 1/4. Simple, right? For the second inequality, x > 2, draw another number line. Mark 2 on the number line. Since x is greater than 2, we'll use an open circle at 2. Then, shade the number line to the right of 2, representing all the values of 'x' that are greater than 2. Number lines are super helpful because they give you a clear picture of all the possible solutions. It's like a visual cheat sheet for inequalities.

Imagine the number line as a road. In our first inequality, we have an open circle in the 1/4 position. It means that the road starts right before the 1/4 position. If our inequality was like x ≤ 1/4, in this case, we would have a close circle at the 1/4 position. It means that 1/4 is part of our possible values.

Key Takeaways and Tips for Success

So, what are the key takeaways from all of this? First, remember the basic steps: isolate the variable, combine like terms, and perform operations on both sides of the inequality. Second, the golden rule: flip the inequality sign when multiplying or dividing by a negative number. Third, number lines are your friends! They help you visualize the solutions. Here's a quick recap of the tips:

• Practice, practice, practice: The more problems you solve, the more comfortable you'll become. Do as many examples as you can.
• Show your work: Write out each step clearly. This helps you avoid mistakes and makes it easier to track down errors if you make them.
• Double-check your answers: Plug your solutions back into the original inequality to make sure they're correct. It's an easy way to catch mistakes.
• Use Number lines: It helps you check and understand your answer

Inequalities might seem tricky at first, but with practice and these tips, you'll be solving them like a pro in no time! Remember to focus on the basics, be careful with negative numbers, and always double-check your work. You've got this!

Further Exploration

Ready to take it to the next level? Here are some ideas for further exploration:

• Compound Inequalities: These involve two or more inequalities joined by